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Full text of "The History of the Calculus and its Conceptual Development"

E HISTORY 
OF THE 
CALCULOS 



(THE CONCEPTS OF THE CALCULUS) 










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(continued on back flap} 



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The History of the Calculus 

and its 
Conceptual Development 

(The Concepts of the Calculus) 

By 
CARL B. BOYER 



with a Foreword by 
RICHARD COURANT 



DOVER PUBLICATIONS, INC., NEW YORK 



Copyright © 1949 by Carl B. Boyer. 
All rights reserved under Pan American and In- 
ternational Copyright Conventions, 



Published in Canada by General Publishing Com- 
pany, Ltd., 30 Lesmill Road, Don Mills, Toronto, 
Ontario, 

Published in the United Kingdom by Constable 
and Company, Ltd,. 10 Orange Street, London 
WC2. 



This Dover edition, first published in 1959, is an 
unabridged and unaltered republication of the work 
originally published by Hafncr Publishing Company, 
Inc., in 1949 under the title The Concepts of the. 
Calculus, A Critical and Historical Discussion of the 
Derivative and the Integral. 



Standard Book Number; -f 86-60509-4 
Library of Congress Catalog Card Number: 59-9673 

Manufactured in the United States of America 
Dover Publications, Inc. 

180 Variek Street 
New York, N, Y. 10014 



Foreword 

Differential and Integral Calculus and Mathematical 
Analysis in general is one of the great achievements of the 
human mind. Its place between the natural and humanistic sciences 
should make it a singularly fruitful medium of higher education. 
Unfortunately, the mechanical way in which calculus sometimes is 
taught fails to present the subject as the outcome of a dramatic 
intellectual struggle which has lasted for twenty-five hundred years 
or more, which is deeply rooted in many phases of human endeavors 
and which will continue as long as man strives to understand himself 
as well as nature. Teachers, students, and scholars who really want 
to comprehend the forces and appearances of science must have some 
understanding of the present aspect of knowledge as a result of his- 
torical evolution. As a matter of fact, reaction against dogmatism 
in scientific teaching has aroused a growing interest in history of 
science; during the recent decades very great progress has been made 
in tracing the historical roots of science in general and mathematics 
in particular. 

The present volume, which fortunately can appear in a second 
printing, is an important contribution towards clarification of the 
many steps which led to the development of the concepts of calculus 
from antiquity to the present day; beyond that, it gives a connected 
and highly readable account of this fascinating story. The book 
ought to reach every teacher of mathematics; then it certainly will 
have a strong influence towards a healthy reform in the teaching of 
mathematics, 

R. COUKANT 

Chairman of the Mathematics Department 
Graduate School, New York University 



Preface 

SOME ten years ago Professor Frederick Barry, of Columbia 
University, pointed out to me that the history of the calculus 
had not been satisfactorily written. Other duties and inadequate 
preparation at the time made it impossible to act upon this sugges- 
tion, but my studies of the past several years have confirmed this view. 
There is indeed no lack of material on the origin and subject matter 
of the calculus, as the titles in the bibliography appended to this work 
will attest. What is wanting is a satisfactory critical account of the 
filiation of the fundamental ideas of the subject from their incipiency 
in antiquity to the final formulation of these in the precise concepts 
familiar to every student of the elements of modern mathematical 
analysis. The present work is an attempt to supply, in some measure, 
this deficiency. An authoritative and comprehensive treatment of the 
whole history of the elementary calculus is greatly to be desired; but 
any such ambitious project is far beyond the scope and intention of the 
dissertation here presented. This is not a history of the calculus in 
all its aspects, but a suggestive outline of the development of the 
basic concepts, and as such should be of service both to students of 
mathematics and to scholars in the field of the history of thought. 
The aim throughout has therefore been to secure clarity of exposition, 
rather than to present a confusingly elaborate all-inclusiveness of 
detail or to display a meticulously precise erudition. This has neces- 
sitated a judicious selection and presentation of such material as 
would preserve the continuity of thought, but it is to be hoped that 
historical accuracy and perspective have not thereby been sacrificed. 
The inclusion at the end of this volume of an extensive bibliography 
of works to which reference has been made has caused it to appear 
unnecessary to give full citations in the footnotes. In these notes 
author and title— in some cases abbreviated— alone have been given; 
titles of books have been italicized, those of articles in the periodical 
literature appear in Roman type enclosed within quotation marks. It is 
felt that the inclusion of such a list of sources on the subject may serve 
to encourage further investigations into the history of the calculus. 



Pref 



ace 



The inspiration toward the projection and completion of the present 
study has been due to Professor Barry, who has generously assisted 
in its prosecution through advice based on his wide familiarity with 
the field of the history of science. Professor Lynn Thorndike, of 
Columbia University, very kindly read and offered his competent 
criticism of the chapter on "Medieval Contributions." Professors L. 
P. SicelofT, of Columbia University, L. C. Karpinski, of the University 
of Michigan, and H. F. MacNeish, of Brooklyn College, have also 
read the manuscript and have furnished valuable aid and suggestions, 
Mrs. Boyer has been unstinting in her encouragement and promotion 
of the work, and has painstakingly done all of the typing. The com- 
position of the Index has been undertaken by the Columbia Univer- 
sity Press. Finally, from the American Council of Learned Societies 
came the subvention, in the form of a grant in aid of publication, 
which has made possible the appearance of this book in its present 
form. To all who have thus contributed toward the preparation and 
publication of this volume I wish to express my sincere appreciation. 

Cakl B. Boyer 
Brooklyn College 
January 3, 1939 



Preface to the Second Printing 

It is gratifying to find sufficient demand for a work on the history 
of the calculus to warrant a second printing. This appears to be 
a token of the increasing awareness in academic circles of the need 
for a broad outlook with respect to science and mathematics. Amaz- 
ing achievements in technology notwithstanding, there is a keener 
appreciation of the fact that science is a habit of mind as well as a 
way of life, and that mathematics is an aspect of culture as well as a 
collection of algorithms. The history of these subjects can never be 
a substitute for work in the laboratory or for training in techniques, 
but it can serve effectively to alleviate the lack of mutual understand- 
ing too often existing between the humanities and the sciences. Per- 
haps even more important is the role that the history of mathematics 
and science can play in the cultivation among professional workers 
in the fields of a sense of proportion with respect to their subjects. 
No scholar familiar with the historical background of his specialty is 
likely to succumb to that specious sense of finality which the novitiate 
all too frequently experiences. For this reason, if for no other, it 
would be wise for every prospective teacher to know not only the 
material of his field but also the story of its development. 

In this new printing a few minor errors in the text have been cor- 
rected. Were it a new edition, more extensive alterations might have 
been justified. These would not have changed substantially the 
general account or the point of view; but the argument would have 
been clarified along the lines suggested by the judicious reviews of 
Julio Rey Pastor (ArcheloiK XXIII [1940] 199-203), L B. Cohen (I sis 
XXXII 11940] 205-210), and others. Additional bibliographical 
references could have been added, of which one in particular deserves 
to be noted here — G. Castelnuovo, Lc origini del calcolo infinites*- 
male neW era moderna (Bologna, 1938). Castelnuovo's book, which 
appeared at about the same time as the present work, should be 
consulted for further details on the modern period written by a 
celebrated geometer. 
The author has been engaged for the past several years in the 



Preface 

preparation of a sort of companion volume on the history of analytic 
geometry. The manuscript of this work has been completed and 
the book should appear before long under the auspices of Scripta 
Matkematica. 

The reappearance of The Concepts of tite Calcidus> which has been 
out of print for well over half a dozen years, is due to Herbert Axel rod 
and Martin N. Wright; and the author wishes to express his appre- 
ciation of their initiative in making the republication possible. And 
to Professor Richard Courant the writer acknowledges a debt of 
gratitude in view of the fact that he graciously consented to write a 
foreword for the new printing. 

Cael B. Boyer 
January 27, 1949 



Contents 



1 



I. INTRODUCTION 

II, CONCEPTIONS IN ANTIQUITY 14 

III. MEDIEVAL CONTRIBUTIONS 61 

IV. A CENTURY OF ANTICIPATION 96 
V. NEWTON AND LEIBNIZ 187 

VI. THE PERIOD OF INDECISION 224 

VII. THE RIGOROUS FORMULATION 267 

VIII. CONCLUSION 2 " 

BIBLIOGRAPHY 311 

INDEX 337 



The History of the Calculus 



I. Introduction 

MATHEMATICS has been an integral part of man's intellectual 
training and heritage for at least twenty-five hundred years. 
During this long period of time, however, no general agreement has 
been reached as to the nature of the subject, nor has any universally 
acceptable definition been given for it. 1 

From the observation of nature, the ancient Babylonians and Egyp- 
tians built up a body of mathematical knowledge which they used in 
making further observations. Thales perhaps introduced deductive 
methods; certainly the mathematics of the early Pythagoreans was 
deductive in character. The Pythagoreans and Plato 2 noted that the 
conclusions they reached deductively agreed to a remarkable extent 
with the results of observation and inductive inference. Unable to 
account otherwise for this agreement, they were led to regard mathe- 
matics as the study of ultimate, eternal reality, immanent in nature 
and the universe, rather than as a branch of logic or a tool of science 
and technology. An understanding of mathematical principles, they 
decided, must precede any valid interpretation of experience. This 
view is reflected in the Pythagorean dictum that all is number, 3 and 
in the assertion attributed to Plato that God always plays the 
geometer. 4 

Later Greek skeptics, it is true, questioned the possibility of 
attaining any knowledge of such absolute character either by reason 
or by experience. But Aristotelian science had meanwhile shown that 
through observation and logic one can at least reach a consistent 
representation of phenomena, and mathematics consequently became; 
with Euclid, an idealized pattern of deductive relationships. Derived 
from postulates consistent with the results of induction from observa- 
tion, it was found serviceable in the interpretation of nature. 

i Bell, The Queen of the Sciences, p. 15. For full citations of works referred to in the 
footnotes see the Bibliography. 

* See, for example, Republic VII, 527, in Dialogues, Jowett trans., Vol. II, pp. 362-63. 
» See Aristotle, Metaphysics 987a-989b, in Works, ed. by Ross and Smith, Vol. VIII; 

Cf. also 1090a. 

* Plutarch, Miscellanies and Essays, III, 402. 



2 Introduction 

The Scholastic view, which prevailed during the Middle Ages, was 
that the universe is "tidy" and simply intelligible. In the fourteenth 
century came a fairly clear realization that Peripatetic qualitative 
views of motion and variation could better be replaced by quantitative 
study. These two concepts, with a revival of interest in Platonic views, 
brought about in the fifteenth and sixteenth centuries a renewal of the 
conviction that mathematics is in some way independent of, and prior 
to, experiential and intuitive knowledge. Such conviction is marked in 
the thinking of Nicholas of Cusa, Kepler, and Galileo, and to a certain 
extent appears in that of Leonardo da Vinci. 

This conception of mathematics as the basis of the architecture 
of the universe was in turn modified in the sixteenth and seventeenth 
centuries. In mathematics, the cause of the change was the less critical 
and more practical use of the algebra which had been adopted from the 
Arabs, early in the thirteenth century, and then further developed in 
Italy. In natural science, the change was due to the rise of experimental 
method. The certitude in mathematics of which Descartes, Boyle, and 
others spoke was thus interpreted to mean a consistency to be found 
rather in the character of its reasoning than in any ontological neces- 
sity which it presented a priori. 

The centering of attention on the procedures rather than on the 
bases of mathematics was emphasized in the eighteenth century by 
an extraordinary success in applying the calculus to scientific and 
mathematical problems. A more critical attitude was inaugurated in 
the nineteenth century by persistent efforts to find a satisfactory 
foundation for the conceptions involved in this new analysis of the 
infinite. Mathematical rigor was revived, and it was discovered that 
Euclid's postulates are not categorical synthetic judgments, as Kant 
maintained, 6 but simply assumptions. Such premises, it was found, 
may be so freely and arbitrarily chosen that — subject to the condition 
that they be mutually compatible — they may be allowed to contra- 
dict the apparent evidence of the senses. Toward the close of the 
century, as the result of the arithmetizing tendency in mathematical 
analysis, it was further discovered that the concept of infinity, tran- 
scending all intuition and analysis, could be introduced into mathe- 
matics without impairing the logical consistency of the subject. 

6 See Sammtliche Werke, II, passim. 



Introduction 3 

If the assumptions of mathematics are quite independent of the 
world of the senses, and if its elements transcend all experience, 6 
the subject is at best reduced to bare formal logic and at worst to 
symbolical tautologies. The formal symbolic and arithmetizing tend- 
ency in mathematics has met with remarkable success in the study 
of the continuous. It has also led to stubborn paradoxes, a fact which 
has aroused increased interest in the nature of mathematics: its scope 
and place in intellectual life, the psychological source of its elements 
and postulates, the logical force of its propositions and their validity 
as interpretations of the world of sense perception. 

The old idea that mathematics is the science of quantity, or of 
space and number, has largely disappeared. The untutored intuition 
of space, it is realized, leads to contradictions, a fact which upsets the 
Kantian view of the postulates. Nevertheless the mathematician is 
guided, although he is not controlled, by the external world of sense 
perception. 7 The mathematical theory of continuity originated in 
direct experience, but the definition of the continuum adopted in the 
end by the mathematician transcends sensory imagination. From this, 
mathematical formalists conclude that since we make no use of intui- 
tion in mathematical definitions and premises, it is not necessary that 
we should interpret the axioms or have any idea as to the nature of 
the objects and relations involved. The intuitionists, however, insist 
that the mathematical symbols involved should significantly express 
thoughts. 8 Although there are two (or more) views of the grounds for 
believing in the unassailable exactness of mathematical laws, the 
recognition that mathematical concepts are suggested, although not 
defined, by intuition thus easily accounts for the fact that the results 
of mathematical deductive reasoning are in apparent agreement with 
those of inductive experience. The derivative and the integral had 
their sources in two of the most obvious aspects of nature — multi- 
plicity and variability— but were in the end defined as mathematical 
abstractions based on the fundamental concept of the limit of an 

•Bertrand Russell has taken advantage of this disconcerting situation to define 
mathematics facetiously as "the subject in which we never know what we are talking 
about, nor whether what we are saying is true." "Recent Work on the Principles of 
Mathematics," p. 84. 

7 B6cher, "The Fundamental Concepts and Methods of Mathematics. 

■ Brouwer, "Intuitionism and Formalism." 



4 Introduction 

infinite sequence of elements. Once we have traced this development, 
the power and fecundity of these ideas when applied to the interpreta- 
tion of nature will be easily understood. 

The calculus had its origin in the logical difficulties encountered 
by the ancient Greek mathematicians in their attempt to express 
their intuitive ideas on the ratios or proportionalities of lines, which 
they vaguely recognized as continuous, in terms of numbers, which they 
regarded as discrete. It became involved almost immediately with 
the logically unsatisfactory (but intuitively attractive) concept of the 
infinitesimal. Greek rigor of thought, however, excluded the infinitely 
small from geometrical demonstrations and substituted the circum- 
ventive but cumbersome method of exhaustion. Problems of variation 
were not attacked quantitatively by Greek scientists. No method 
could be developed which would do for kinematics what the method 
of exhaustion had done for geometry— indicate an escape from the 
difficulties illustrated by the paradoxes of Zeno. The quantitative 
study of variability, however, was undertaken in the fourteenth cen- 
tury by the Scholastic philosophers. Their approach was largely 
dialectical, but they had resort as well to graphical demonstration. 
This method of study made possible in the seventeenth century the 
introduction of analytic geometry and the systematic representation 
of variable quantities. 

The application of this new type of analysis, together with the free 
use of the suggestive infinitesimal and the more extensive application 
of numerical concepts, led within a short time to the algorithms of 
Newton and Leibniz, which constitute the calculus. Even at this 
stage, however, there was no clear conception of the logical basis of 
the subject. The eighteenth century strove to find such a basis, and 
although it met with little success in this respect, it did in the effort 
largely free the calculus from intuitions of continuous motion and 
geometrical magnitude. Early in the following century the concept 
of the derivative was made fundamental, and with the rigorous defi- 
nitions of number and of the continuum laid down in the latter half 
of the century, a sound foundation was completed. Some twenty-five 
hundred years of effort to explain a vague instinctive feeling for con- 
tinuity culminated thus in precise concepts which are logically denned 
but which represent extrapolations beyond the world of sensory ex- 



Introduction 5 

perience. Intuition, or the putative immediate cognition of an ele- 
ment of experience which ostensibly fails of adequate expression, in 
the end gave way, as the result of reflective investigation, to those 
well-defined abstract mental constructs which science and mathematics 
have found so valuable as aids to the economy of thought. 

The fundamental definitions of the calculus, those of the derivative 
and the integral, are now so clearly stated in textbooks on the subject, 
and the operations involving them are so readily mastered, that it is 
easy to forget the difficulty with which these basic concepts have been 
developed. Frequently a clear and adequate understanding of the 
fundamental notions underlying a branch of knowledge has been 
achieved comparatively late in its development. This has never been 
more aptly demonstrated than in the rise of the calculus. The precision 
of statement and the facility of application which the rules of the cal- 
culus early afforded were in a measure responsible for the fact that 
mathematicians were insensible to the delicate subtleties required in 
the logical development of the discipline. They sought to establish the 
calculus in terms of the conceptions found in the traditional geometry 
and algebra which had been developed from spatial intuition. During 
the eighteenth century, however, the inherent difficulty of formulating 
the underlying concepts became increasingly evident, and it then 
became customary to speak of the "metaphysics of the calculus," thus 
implying the inadequacy of mathematics to give a satisfactory exposi- 
tion of the bases. With the clarification of the basic notions— which, 
in the nineteenth century, was given in terms of precise mathematical 
terminology — a safe course was steered between the intuition of the 
concrete in nature (which may lurk in geometry and algebra) and the 
mysticism of imaginative speculation (which may thrive on tran- 
scendental metaphysics). The derivative has throughout its develop- 
ment been thus precariously situated between the scientific phe- 
nomenon of velocity and the philosophical noumenon of motion. 

The history of the integral is similar. On the one hand, it has offered 
ample opportunity for interpretations by positivistic thought in terms 
either of approximations or of the compensation of errors — views 
based on the admitted approximative nature of scientific measure- 
ments and on the accepted doctrine of superimposed effects. On 
the other hand, it has at the same time been regarded by idealistic 



6 Introduction 

metaphysics as a manifestation that beyond the finitism of sensory 
percipiency there is a transcendent infinite which can be but asymp- 
totically approached by human experience and reason. Only the pre- 
cision of their mathematical definition — the work of the past century — 
enables the derivative and the integral to maintain their autonomous 
position as abstract concepts, perhaps derived from, but nevertheless 
independent of, both physical description and metaphysical ex- 
planation. 

At this point it may not be undesirable to discuss these ideas, with 
reference both to the intuitions and speculations from which they were 
derived and to their final rigorous formulation. This may serve to 
bring vividly to mind the precise character of the contemporary con- 
ceptions of the derivative and the integral, and thus to make unam- 
biguously clear the terminus ad quern of the whole development. 

The derivative is the mathematical device used to represent point 

properties of a curve or function. It thus has as its analogues in science 

instantaneous properties of a body in motion, such as the velocity of 

the object at any given time. When science is concerned with a time 

interval, the average velocity over this interval is suitably defined as 

the ratio of the change in the distance covered during the interval 

to the time interval itself. This ratio is conveniently represented by 

As 
the notation — . Inasmuch as the laws of science are formulated by 
At 

induction on the basis of the evidence of the senses, on the face of it 
there can be no such thing in science as an instantaneous velocity, 
that is, one in which the distance and time intervals are zero. The 
senses are unable to perceive, and science is consequently unable to 
measure, any but actual changes in position and time. The power of 
every sense organ is limited by a minimum of possible perception. 9 
We cannot, therefore, speak of motion or velocity, in the sense of a 
scientific observation, when either the distance or the corresponding 
time interval becomes so small that the minimum of sensation in- 
volved in its measurement is not excited — much less when the interval 
is assumed to be zero. 

If, on the other hand, the distance covered is regarded as a function 
of the time elapsed, and if this relationship is represented mathe- 

• Cf. Mach, Die Principien der WSrmelekre, pp. 71-77. 



Introduction 7 

matically by the equation 5 = /(/), the minimum of sensation no 
longer operates against the consideration of the abstract difference 

AS 

quotient — . This has a mathematical meaning no matter how small 
At 

the time and distance intervals may be, provided, of course, that the 
time interval is not zero. Mathematics knows no minimum interval 
of continuous magnitudes — and distance and time may be considered 
as such, inasmuch as there is no evidence which would lead one to 
regard them otherwise. Attempts to supply a logical definition of such 
an infinitesimal minimum which shall be consistent with the body of 
mathematics as a whole have failed. Nevertheless, the term "instan- 
taneous velocity" appears to imply that the time interval is to be 
regarded not only as arbitrarily small but as actually zero. Thus the 
term predicates the very case which mathematics is compelled to 
exclude because of the impossibility of division by zero. 

This difficulty has been resolved by the introduction of the deriva- 
tive, a concept based on the idea of the limit. In considering the 

As 
successive values of the difference quotient — , mathematics may con- 

At 

tinue indefinitely to make the intervals as small as it pleases. In this 

way an infinite sequence of values, r h r 2 , r 3 , . . . , r n , . . . (the suc- 

As\ 
cessive values of the ratio — J is obtained. This sequence may be such 

At) 

that the smaller the intervals, the nearer the ratio r n will approach 

to some fixed value L, and such that by taking the value of n to be 

sufficiently large, the difference \L— r n \ can be made arbitrarily 

small. If this be the case, this value L is said to be the limit of the 

infinite sequence, or the derivative /'(/) of the distance function 

f(i), or the instantaneous velocity of the body. It is to be borne in 

mind, however, that this is not a velocity in the ordinary sense and 

has no counterpart in the world of nature, in which there can be no 

motion without a change of position. The instantaneous velocity as 

thus defined is not the division of a time interval into a distance 

ds 
interval, howsoever much the conventional notation — = f'(t) may 

at 

suggest a ratio. This symbolism, although remarkably serviceable in 



8 Introduction 

the carrying out of the operations of the calculus, will be found to 
have resulted from misapprehension on the part of Leibniz as to the 
logical basis of the calculus. 

The derivative is thus denned not in terms of the ordinary proc- 
esses of algebra, but by an extension of these to include the concept 
of the limit of an infinite sequence. Although science may not extra- 
polate beyond experience in thus making the intervals indefinitely 
small, and although such a process may be "inadequately adapted to 
nature," 10 mathematics is at liberty to introduce the new limit con- 
cept, on the basis of the logical definition given above. One can, of 
course, make this notion still more precise by eliminating the words 
"approach," "sufficiently large," and "arbitrarily small," as follows: 
L is said to be the limit of the above sequence if, given any positive 
number e (howsoever small), a positive integer N can be found 
such that for n > N the inequality \L — r n \ < e is satisfied. 

In this definition no attempt is made to determine any so-called 
"end" of the infinite sequence, or to deal with the possibility that the 
variable r n may "reach" its limit L. The number L, thus abstractly 
defined as the derivative, is not to be regarded as an "ultimate ratio," 
nor may it be invoked as a means of "visualizing" an instantaneous 
velocity or of explaining in a scientific or a metaphysical sense either 
motion or the generation of continuous magnitudes. It is such unclear 
considerations and unwarranted interpretations which, as we shall 
see, have embroiled mathematicians, since the time of Zeno and his 
paradoxes, in controversies which often misdirected their energy. On 
the other hand, however, it is precisely such suggestive notions which 
stimulated the investigations resulting in the formal elaboration of the 
calculus, even though this very elaboration was in the end to exclude 
them as logically irrelevant. 

Just as the problem of defining instantaneous velocities in terms of 
the approximation of average velocities was to lead to the definition 
of the derivative, so that of defining lengths, areas, and volumes of 
curvilinear configurations was to eventuate in the formulation of the 
definite integral. This concept, however, was likewise ultimately to 
be so defined that the geometrical intuition which gave it birth was 
excluded. As part of the Qreek pursuit of unity in multiplicity, we 

10 Schrodinger, Science and the Human Temperament, pp. 61-62. 



Introduction 9 

shall see that an attempt was made to inscribe successively within a 
circle polygons of a greater and greater number of sides in the hope 
of finally "exhausting" the area of the circle, that is, of securing a 
polygon with so great a number of sides that its area would be equal 
to that of the circle. This naive attempt was of course doomed to 
failure. The same process, however, was adopted by mathematicians 
as basic in the definition of the area of the circle as the limit A of the 
infinite sequence formed by the areas A h At, A 3 , . . . , A n , ... of 
the approximating polygons. This affords another example of extra- 
polating beyond sensory intuition, inasmuch as there is no process 
by which the transition from the sequence of polygonal areas to the 
limiting area of the circle can be "visualized." An infinite subdivision 
is of course excluded from the realm of sensory experience by the fact 
that there exist thresholds of sensation. It must be banished also from 
the sphere of thought, in the physiological sense, inasmuch as psychol- 
ogy has shown that for an act of thought a measurable minimum of 
duration of time is required. 11 Logical definition alone remains a 
sufficient criterion for the validity of this limiting value A . 

In order to free the limiting process just described from the geo- 
metrical intuition inherent in the notion of area, mathematics was 
constrained to give formal definition to a concept which should not 
refer to the sense experience from which it had arisen. (This followed 
a long period of indecision, the course of which we are shortly to 
trace.) After the introduction of analytical geometry it became cus- 
tomary, in order to find the area of a curvilinear figure, to substitute 
for the series of approximating polygons a sequence of sums of approxi- 
mating rectangles, as illustrated in the diagram (fig. 1). The area of 
each of the rectangles could be represented by the notation /(#,) A*,- 

n 

and the sum of these by the symbolism S n = 2 /(*,-) A*,-. The area 

of the figure could then be defined as the limit of the infinite sequence 
of sums S n , as the number of subdivisions n increased indefinitely and 
as the intervals Ax { approached zero. Having set up the area in this 
manner with the help of the analytical representation of the curve, it 
then became a simple matter to discard the geometrical intuition 
leading to the formation of these sums and to define the definite 

u Enriques, Problems of Science, p. 15. 



10 



Introduction 



integral of /(#) over the interval from x = a to x = b arithmetically 

n 

as the limit of the infinite sequence of sums S n = 2 /(#.) A#; (where 

the divisions A*,- are taken to cover the interval from a to b) as the 
intervals Ax f become indefinitely small. This definition then invokes, 
apart from the ordinary operations of arithmetic, only the concept of 
the limit of an infinite sequence of terms, precisely as does that of the 
derivative. The realization of this fact, however, followed only after 
many centuries of investigation by mathematicians. The very nota- 
tion i a f(x)dx, which is now customarily employed to represent this 
definite integral, is again the result of the historical development of 
the concept rather than of an effort to represent the final logical 




FIGURE 1 

formulation. It is suggestive of a sum, rather than of the limit of an 
infinite sequence; and in this respect it is in better accord with the 
views of Leibniz than with the intention of the modern definition. 

The definite integral is thus defined independently of the derivative, 
but Newton and Leibniz discovered the remarkable property con- 
stituting what is commonly known as the fundamental theorem of the 
calculus, viz., that the definite integral F(x) = Jlf(x)dx of the con- 
tinuous function f(x) has a derivative which is this very same func- 
tion, F'{x) = f(x). That is, the value of the definite integral of f(x) 
from a to b may in general be found from the values, for x — a and 
x = b, of the function F(x) of which f(x) is the derivative. This rela- 
tionship between the derivative and the definite integral has been 
called "the root idea of the whole of the differential and integral 



Introduction 11 

calculus." 12 The function F(x), when so denned, is often called the 
indefinite integral of /(*), but it is to be recognized that it is in this 
case not a numerical limit given by an infinite sequence, as is the 
definite integral, but is a function of which f(x) is the derivative. 

The function F(x) is sometimes also called the primitive of f{x), 
and the value of F(b) — F(a) is occasionally taken as the definition 

of Sif(x)dx. In this case the relationship ! b J(x)dx = Urn 2 /(*,) Ax, is 

then the fundamental theorem of the calculus, rather than the defi- 
nition of the definite integral. 

Although the recognition of this striking inverse relationship, 
together with the formulation of rules of procedure, may be taken as 
constituting the invention of the subject, it is not to be supposed that 
the inventors of the calculus were in possession of the above sophis- 
ticated concepts of the derivative and the integral, so necessary in 
the logical development of the new analysis. More than a hundred 
years of investigation was to be required before the achievement of 
their final definition in the nineteenth century. 

It is the purpose of this essay to trace the development of these 
two concepts from their incipiency in sense experience to their final 
elaboration as mathematical abstractions, defined in terms of formal 
logic by means of the idea of the limit of an infinite sequence. We 
shall find that the history of the calculus affords an unusually striking 
example of the slow formation of mathematical concepts by the 
emancipation from all sense data of ideas born of our primary intui- 
tions. The derivative and the integral are, in the last analysis, syn- 
thetically defined in terms of ordinal considerations and not of those 
of continuous quantity and variability. They are, nevertheless, the 
results of attempts to schematize our sense impressions of these last 
notions. This explains why the calculus, in the early stages of its de- 
velopment, was bound up with concepts of geometry or motion, and 
with explanations of indivisibles and the infinitely small; for these 
ideas are suggested by naive intuition and experience of continuity. 

There is in the calculus a further concept which merits brief con- 
sideration at this point, not so much on account of any logical exigen- 
cies in the present structure of the calculus as to make clearer its 
historical development. The infinite sequences considered above in 

u Courant, Differential and Integral Calculus, II, 111. 



12 Introduction 

the definitions of the derivative and of the integral were obtained by- 
continuing, in thought, to diminish ad infinitum the intervals between 
the values of the independent variable. Considerations which in 
physical science led to the atomic theory were at various periods in the 
development of the calculus adduced in mathematics. These made it 
appear probable that just as in the actual subdivision of matter 
(which has the appearance of being continuous) we arrive at ultimate 
particles or atoms, so also in continuous mathematical magnitudes we 
may expect (by means of successive subdivisions carried on in thought) 
to obtain the smallest possible intervals or differentials. The derivative 
would in this case be defined as the quotient of two such differentials, 
and the integral would then be the sum of a number (perhaps finite, 
perhaps infinite) of such differentials. 

There is, to be sure, nothing intuitively unreasonable in such a 
view; but the criterion of mathematical acceptability is logical self- 
consistency, rather than reasonableness of conception. Such a view of 
the nature of the differential, although possessing heuristic value in 
the application of the calculus to problems in science, has been judged 
inacceptable in mathematics because no satisfactory definition has as 
yet been framed which is consistent with the principles of the calculus 
as formulated above, or which may be made the basis of a logically 
satisfactory alternative exposition. In order to retain the operational 
facility which the differential point of view affords, the concept of the 
differential has been logically defined, not in terms of mathematical 
atomism, but as a notion derived from that of the derivative. The 
differential dx of an independent variable x is to be thought of as 
nothing but another independent variable; but the differential dy of 
a function y = f(x) is defined as that variable the values of which 
are so determined that for any given value of the variable dx the 

. dy 
ratio — shall be equal to the value of the derivative at the point in 
dx 

question, i. e., dy = f(x)dx. The differentials as thus defined are only 
new variables, and not fixed infinitesimals, or indivisibles, or "ulti- 
mate differences," or "quantities smaller than any given quantity," 
or "qualitative zeros," 13 or "ghosts of departed quantities," as they 

u "Abominable little zeroes," they have been called by Osgood. See Osgood, "The 
Calculus in Our Colleges and Technical Schools"; and Huntington, "Modern Interpreta- 
tion of Differentials." 



Introduction 13 

have been variously considered in the development of the cal- 
culus. 

Poincare has said that had mathematicians been left the prey of 
abstract logic, they would never have gotten beyond the theory of 
numbers and the postulates of geometry. 14 It was nature which thrust 
upon mathematicians the problems of the continuum and of the cal- 
culus. It is therefore quite understandable that the persistent atom- 
istic speculations of physical thought should have had a counterpart 
in attempts to picture, by means of indivisible elements, the space 
described by geometry. The further development of mathematics, 
however, has shown that such notions must be abandoned, in order to 
preserve the logical consistency of the subject. The basis of the con- 
cepts leading to the derivative and the integral was first found in 
geometry, for despite the apodictic character of its proofs, this sub- 
ject was considered an abstract idealization of the world of the senses. 

Recently, however, it has been more clearly perceived that mathe- 
matics is the study of relationships in general and must not be ham- 
pered by any preconceived notions, derived from sensory perception, 
of what these relationships should be. The calculus has therefore been 
gradually emancipated from geometry and has been made dependent, 
through the definitions of the derivative and the integral, on the 
notion of the natural numbers, an idea from which all traditional pure 
mathematics, including geometry, can be derived. 15 Mathematicians 
now feel that the theory of aggregates has provided the requisite 
foundations for the calculus, for which men had sought since the time 
of Newton and Leibniz. 16 It is impossible to predict with any con- 
fidence, however, that this is the final step in the process of abstracting 
from the primitive ideas of change and multiplicity all those irrelevant 
incumbrances with which intuition binds these concepts. It is a 
natural tendency of man to hypostatize those ideas which have great 
value for him, 17 but a just appreciation of the origin of the derivative 
and the integral will make clear how unwarrantedly sanguine is any 
view which would regard the establishment of these notions as bringing 
to its ultimate close the development of the concepts of the calculus. 

14 See Osgood, op. cit., p. 457; Poincar6, Foundations of Science, p. 46. 

16 Russell, Introduction to Mathematical Philosophy, p. 4. See also Poincare, Foundations 
of Science, pp. 441, 462. 

w Russell, The Principles of Mathematics, pp. 325-26. 

17 Mach, The Science of Mechanics, p. 541. 



II. Conceptions in Antiquity 

THE PRE-HELLENIC peoples are usually regarded as pre- 
scientific in their attitude toward nature, 1 inasmuch as they pal- 
pably lacked the Greek confidence in its essential reasonableness, as 
well as the associated feeling that beneath the perplexing hetero- 
geneity and ceaseless flux of events would be found elements of 
uniformity and permanence. 

The search for universals, which the Greeks maintained so per- 
sistently, apparently held no attraction for the Egyptians and the 
Babylonians. So also the mathematical thought of these peoples — 
about whom, among those of the ancient world, we are best informed — 
bore no significant resemblance to ours, in that it lacked the tendency, 
essential to both mathematical and scientific method, toward the 
isolation and abstraction of certain samenesses from their confusingly 
varied concomitants in nature and in thought. Lacking these ele- 
ments of invariance to serve as premises of inference, they were 
accordingly without appreciation of the characteristics which dis- 
tinguish mathematics from science, namely, its logical nature and the 
necessity of deductive proofs. 2 

A large body of knowledge of spatial and numerical relations they 
did, however, acquire; and the more familiar their work becomes, the 
more it inspires our admiration. 3 It was, however, largely the result 
of empirical investigations, or at best of generalizations which were 
the result of incomplete induction from simple to more complicated 
cases. The Egyptian rule for computing the volume of a square 
pyramid — from which was obtained the most remarkable of all 

1 On this point, however, there are significant differences of opinion. Barry (The 
Scientific Habit of Thought, p. 104) places "the childhood of science" among the early 
Greeks; Burnet (Greek Philosophy, Part I, "Thales to Plato," pp. 4-5) says "natural science 
is the creation of the Greeks," and finds "not the slightest trace of that science in Egypt 
or even in Babylon." On the other hand, Karpinski ("Is There Progress in Mathematical 
Discovery," pp. 51-52) would regard the achievements of the Babylonians, Egyptians, and 
Hindus as "scientific in the highest sense." 

2 Milhaud, Nouvettes etudes sur I'histoire de la pens6e scientifique, pp. 41-133. 

* See Neugebauer, Vorlesungen iiber Gesckichte der antiken mathematischen Wissenschaf- 
ten, Vol. I, Vorgriechische Mathematik, for the best account of this work. 



Conceptions in Antiquity 15 

Egyptian results, the rule for determining the volume of a frustum 
of a square pyramid — was probably the result of this method of pro- 
cedure. 4 That the demonstration could not be correct, in our under- 
standing of the mathematical implications of this term, is clear from 
the fact that this result for the general case requires the use of infini- 
tesimal or limit considerations 5 which, while constituting the point of 
departure for the story of the derivative and integral, are not found 
in any record before the Greek period. 6 

More fundamental than this lack of deductive proofs of inferred 
results is the fact that in all this Egyptian work the rules were applied 
to concrete cases with definite numbers only. 7 There was no con- 
ception in their geometry of a triangle as representative of all tri- 
angles, 8 an abstract generalization necessary for the elaboration of a 
deductive system. This lack of freedom and imagination is apparent 
also in Egyptian arithmetic, into which the abstract number con- 
cept, as such, did not enter, 9 and in which, with the exception of $, 
all rational fractions were expressed as sums of unit fractions. 10 

Babylonian mathematics resembles the Egyptian more than the 
Greek, but with a stronger emphasis on the numerical side and thus 
a more highly developed algebra than that of Egypt. Here again we 
must not look for logical structure or proof, more complicated cases 
being reduced to simpler, and so "proved," or rather treated as 
analogous without proof; 11 and we must remember that this work, like 
the Egyptian, deals with concrete cases only. In connection with 
Babylonian astronomy, we find that problems involving continuous 
variation were studied, but only to the extent of tabulating the values 
of a function (such as the brightness of the moon, for example) for 

* Ibid., p. 128. 

5 See Dehn, "Ueber raumgleiche Polyeder," for proof that infinitesimal considerations 
cannot here be avoided. 

6 Neugebauer, op. cit., pp. 126-28. There seems to be no basis for the implication made 
by Bell (The Search for Truth, p. 191) that the Egyptians used infinite or infinitesimal 
considerations in deductive reasoning. 

7 Neugebauer, op. cit., p. 127. 

8 Luckey, "Was ist agyptische Geometrie?" p. 49. 

9 Neugebauer, op. cit., p. 203. See also Miller, "Mathematical Weakness of the Early 
Civilizations." 

10 For example, the Egyptians did not regard as a single number that number which we 
now represent by the symbol f , but thought of it as the sum of the two fractions, £ and fa 

11 Neugebauer, op. cit., pp. 203-4. 



16 Conceptions in Antiquity 

values of the argument (time) measured at equal intervals, and from 
this calculating the maximum (intensity) of the function. 12 The Greeks 
were the first, however, systematically to analyze 13 the idea of contin- 
uous magnitude and to develop concepts leading to the integral and 
the derivative. 

Our information on the history of mathematics in the interval be- 
tween that of the best Egyptian and Babylonian mathematics and 
the early work in Greece is unfortunately fragmentary. That these 
oriental civilizations influenced Greek culture is clear; but the nature 
and extent of their contribution is undetermined. However that may 
be, there is an obvious change in spirit in both science and mathe- 
matics, as these developed in Greece. The human mind was ''dis- 
covered" as something different from the surrounding body of nature 
and capable of discerning similarities in a multiplicity of events, of 
abstracting these from their settings, generalizing them, and deducing 
therefrom other relationships consistent with further experience. It is 
for this reason that we consider mathematical and scientific method 
as originating with the Hellenic race; 14 but to say that Greek mathe- 
matics and science were autochthonous would be to forget the debt 
of subject-matter owed to Egypt and Babylon. 15 It is likely 16 that the 
new outlook of the Hellenes was the result of the flux of civilizations 
occurring at this time, this impressing upon the rising Greek fortunes 
the stamp of numerous cultures. 

Thales is the first Greek mentioned in connection with this "in- 
tellectual revolution," which produced elementary mathematics and 
which was to reveal those difficulties in conception, the study and 
resolution of which were to produce within the next twenty-five 
hundred years the subject which we now call the calculus. He is said 
to have been a great traveler, to have learned geometry from the 
Egyptians and astronomy from the Babylonians, and upon his return 
to Greece to have instructed his successors in the principles of these 

u Hoppe, "Zur Geschichte der Infinitesimalrechnung bis Leibniz und Newton." 

u Neugebauer, op. cit., p. 205. 

" T. L. Heath, A History of Greek Mathematics, I, v, says, "Mathematics in short is a 
Greek science, whatever new developments modern analysis has brought or may bring." 

15 See Karpinski, "Is There Progress in Mathematical Discovery?" pp. 46-47. Cf. also 
Gandz, "The Origin and Development of the Quadratic Equations in Babylonian, Greek 
and Early Arabic Algebra," pp. 542-43. 

14 As Neugebauer suggests (op. cit., p. 203). 



Conceptions in Antiquity 17 

subjects. Proclus says of his method of attack that it was "in some 
cases more general, in other cases more empirical." Thales' demon- 
strations may therefore have appealed to some extent to the evidence 
of the senses, and in fact his theorems were those the truth of which 
one would recognize by the execution of some practical construction. 17 

To Thales, nevertheless, is ascribed the establishment of mathe- 
matics as a deductive discipline. 18 He did not, however, construct a 
body of mathematical knowledge, nor did he apply his method to the 
analysis of the problem of the continuum. These tasks seem to have 
been performed by Pythagoras, the second Greek mathematician of 
whom we have substantial information. According to Proclus, he 
"transformed the study of geometry into a liberal education, examin- 
ing the principles of the science from the beginning and probing the 
theorems in an immaterial and intellectual manner"; 19 but, beyond 
admitting this, it is impossible to ascribe with any degree of certainty 
other mathematical or scientific accomplishments to Pythagoras the 
individual, since never in antiquity could he be distinguished from 
his school and it is hardly possible to do so now. The knowledge 
acquired by the school established by Pythagoras was held to be 
strictly esoteric, with the result that when the general nature of 
Pythagorean thought became apparent, after the death of the founder 
about 500 b. c, it was already impossible to attribute to a single 
member any one contribution. Nevertheless, the process of abstraction 
begun by Thales was evidently completed by this school. 

A new difficulty, however, then entered into Greek thought, for the 
Pythagorean mathematical concepts, abstracted from sense impres- 
sions of nature, were now in turn projected into nature and con- 
sidered to be the structural elements of the universe. 20 Thus the 
Pythagoreans attempted to construct the whole heaven out of numbers, 
the stars being units which were material points. Later they identi- 
fied the regular geometrical solids, with which they were familiar, with 
the different sorts of substances in nature. 21 Geometry was regarded 

17 Paul Tannery, La Gtomttrie grecque, pp. 89 ff. 
» T. L. Heath, History of Greek Mathematics, I, 128. 

19 Ibid., I, 140. Cf. also Moritz Cantor, Vorlesungen fiber Geschichte der Mathemaiik, 
I, 137. 

M See Brunschvicg, Les tiapes de la philosophic mathSmatique, pp. 34 ff. 
n T. L. Heath, History of Greek Mathematics, I, 165. 



18 Conceptions in Antiquity 

by them as immanent in nature, and the idealized concepts of geom- 
etry appeared to them to be realized in the material world. This 
confusion of the abstract and the concrete, of rational conception and 
empirical description, which was characteristic of the whole Pythag- 
orean school and of much later thought, will be found to bear sig- 
nificantly on the development of the concepts of the calculus. It has 
often been inexactly described as mysticism, 22 but such stigmatization 
appears to be somewhat unfair. Pythagorean deduction a priori 
having met with remarkable success in its field, an attempt (unwar- 
ranted, it is now recognized) was made to apply it to the description 
of the world of events, in which Ionian hylozoistic interpretations a 
posteriori had made very little headway. This attack on the problem 
was highly rational and not entirely unsuccessful, even though it was 
an inversion of the scientific procedure, in that it made induction 
secondary to deduction. 

One very important result of the Pythagorean search for unity in 
nature and geometry was the theory of application of areas. This 
originated with the Pythagoreans, if not with Pythagoras himself, 23 
and became fundamental in Greek geometry, in which it later led to 
the method of exhaustion, the Greek equivalent of our integration. 24 
The method of the application of areas enabled them to say of a figure 
bounded by straight lines that it was greater than, equivalent to, or 
less than 25 a second figure. Such a superposition of one area upon 
another constitutes the first step in the attempt to make exactly 
definable the notion of area, in which a unit of area is said to be 
contained in a second area a given number of times. Modern mathe- 
matics has made fundamental the concept of number rather than that 
of congruence, with the result that the word "area" no longer calls 
vividly to mind that comparison of two surfaces which is essential in 
this connection and which was always uppermost in Greek thought. 
Greek mathematicians did not speak of the area of one figure, but of 
the ratio of two surfaces, a definition which could not, because of the 
problem of incommensurability, be made precise before a satisfactory 
concept of number had been developed. Such a concept the Pythag- 

22 See Russell, Our Knowledge of the External World, p. 19. 

23 T. L. Heath, History of Greek Mathematics, I, 150. 

24 It was basic also in the Greek solution of quadratic equations by geometrical algebra. 
28 Our names for the conic sections (ellipse, hyperbola, and parabola) were, incidentally, 

derived from the designations the Pythagoreans used in this connection. 



Conceptions in Antiquity 19 

oreans did not possess. This contribution was not made until the last 
half of the nineteenth century, and it was to furnish, in the last 
analysis, the basis of the whole of the calculus. However, to the Pythag- 
oreans, in all probability, we owe the recognition of the need for some 
such concept — a discovery which may be regarded as the first step, a 
terminus a quo, in the development of the concepts of the calculus. 

The inadequacy of the Pythagorean view of the ratio of magnitudes 
was first made evident to the followers of this school on the application 
of the doctrine, not to areas, but rather to the analogous comparison 
of lines which is presupposed by our notion of length. Such investiga- 
tions led the Pythagoreans to an intensely disconcerting discqvery. 
If the side of a square were to be applied to the diagonal, no common 
measure could be discovered which would express one in terms of the 
other. In other words, these lines were shown to be incommensurable. 
Just when this discovery took place and whether it was made by 
Pythagoras himself, by the early Pythagoreans, or by later members 
of the school are moot points in the history of mathematics. 26 It has 
also been maintained that Pythagoras owed his knowledge of the 
irrational and of the five regular solids, as well as much of his phi- 
losophy, to the Hindus. 27 

The question as to how the incommensurability was discovered or * 
proved is also difficult to answer with any assurance. The rnethod of 
application would suggest as a form of proof the geometrical equivalent 
of the process of finding the greatest common divisor, but there is 
another aspect of Pythagorean thought which points to a different sort 
of reasoning. A prevailing belief in the unity and harmony of nature 
and knowledge had led the Pythagoreans not only to explain different 
aspects of nature by various mathematical abstractions, as already 
suggested, but also to attempt to identify the realms of number and 
magnitude. 28 By the term number, however, the Pythagoreans did 

26 On this question see the following two papers by Heinrich Vogt: "Die Entdeckungs- 
geschichte des Irrationalen nach Plato und anderen Quellen des 4. Jahrhunderts" : "Zur 
Entdeckungsgeschichte des Irrationalen." Vogt concludes that the discovery was made by 
the later Pythagoreans at some time before 410 b. c. T. L. Heath (History of Greek Mathe- 
matics, I, 157) would place it "at a date appreciably earlier than that of Democritus. M 

27 See Schroeder, Pythagoras und die Inder; and Vogt, "Haben die alten Inder den 
Pythagoreischen Lehrsatz und das Irrationale gekannt?" 

28 For a keenly critical account of the significance, from the scientific point of view, of 
the Pythagorean problem of associating the fields of number and magnitude, see Barry, 
The Scientific Habit of Thought, pp. 207 ff. 



20 Conceptions in Antiquity 

not understand the abstraction to which we give this name, but used 
it to designate "a progression of multitude beginning from a unit and 
a regression ending in it." 29 The integers were thus fundamental, 
numbers being collections of units, and, as was the case with their 
geometrical forms, they were immanent in nature, each having a posi- 
tion and occupying a place in space. If geometrical abstractions were 
the elements of actual things, number was the ultimate element of 
these abstractions and thus of physical bodies and of all nature. 30 
This hypostatization of number had led the Pythagoreans to regard a 
line as made up of an integral number of units. This doctrine could not 
be applied to the diagonal of a square, however, for no matter how 
small a unit was chosen as a measure of the sides, the diagonal could 
not be a "progression of multitude" beginning with this unit. The 
proof of this fact, as given by Aristotle (and which possibly is that of 
the Pythagoreans), 31 is based on the distinction between the odd and 
the even, which the Pythagoreans themselves had emphasized. 

The incommensurability of lines remained ever a stumbling block 
for Greek geometry. That it made a strong impression on Greek 
thought is indicated by the story, repeated by Proclus, that the 
Pythagorean who disclosed the fact of incommensurability suffered 
death by shipwreck as a result. It is demonstrated also, and more 
reliably, by the prominence given to the doctrine of irrationals by 
Plato and Euclid. It never occurred to the Greeks to invent an irra- 
tional number 32 to circumvent the difficulty, although they did develop 
as a part of geometry (found, for example, in the tenth book of 
Euclid's Elements) the theory of irrational magnitudes. Failing to gen- 
eralize their number system along the lines suggested later by the 
development of mathematical analysis, the only escape for Greek 
mathematicians in the end was to abandon the Pythagorean attempt 
to identify the realm of number with that of geometry or of con- 
tinuous magnitude. 

The effort to unite the two fields was not given up, however, before 
intuition had sought another way out of the difficulty. If there is no 

» T. L. Heath, History of Greek Mathematics, I, 69-70. 

30 Milhaud, Les Philosopkes geomitres de la Grece, p. 109. 

31 Zeuthen, "Sur l'origine historique de la connaissance des quantites irrationelles." 

32 Stolz, Vorlesungen fiber Allgemeine Arithmetik, I, 94; cf. also Vogt, Der Grenzbegriff in 
der Elementar-mathematik, p. 48. 



Conceptions in Antiquity 21 

finite line segment so small that the diagonal and the side may both 
be expressed in terms of it, may there not be a monad or unit of 
such a nature that an indefinite number of them will be required for 
the diagonal and for the side of the square? 

We do not know definitely whether or not the Pythagoreans them- 
selves invoked the infinitely small. We do know, however, that the 
concept of the infinitesimal had entered into mathematical thought, 
through a doctrine elaborated in the fifth century b. c, as the result 
of Greek speculation concerning the nature of the physical world. 
After the failure of the early Ionian attempts to find a fundamental 
element out of which to construct all things, there arose at Abdera 
the materialistic doctrine of physical atomism, according to which 
there is no one physis, not even a small group of substances of which 
everything is composed. The Abderitic school held that everything, 
even mind and soul, is made up of atoms moving about in the void, 
these atoms being hard indivisible particles, qualitatively alike but of 
countless shapes and sizes, all too small to be perceived by sense 
impressions. 

There is nothing either logically or physically inconsistent in this 
doctrine, which is a crude anticipation of our own chemical thought; 
but the greatest of the Greek atomists, Democritus, did not stop 
here: he was also a mathematician and carried the idea over into 
geometry. As we now know from the Method of Archimedes, which was 
discovered as a palimpsest in 1906, Democritus was the first Greek 
mathematician to determine the volumes of the pyramid and the 
cone. How he derived these results we do not know. The formula for 
the volume of a square pyramid was probably known to the Egyp- 
tians, 33 and Democritus in his travels may have learned of it and 
generalized the result to include all polygonal pyramids. The result 
for the cone would then be a natural inference from the result of 
increasing indefinitely the number of sides in a regular polygon 
forming the base of a pyramid. This explanation would correspond to 
others involving similar infinitesimal conceptions, which we know 
Democritus entertained and which later influenced Plato. 34 

33 Neugebauer, Vorlesungen iiber Geschichte der Antiken mathematischen Wissenschaften, 
p. 128. 

34 See Luria, "Die Infinitesimaltheorie der antiken Atomisten." 



22 Conceptions in Antiquity 

Aristotle and Euclid ascribe to him a mathematical atomism, and 
we know from Plutarch 35 that he was puzzled as to whether the 
infinitesimal parallel circular sections of which the cone may be con- 
sidered to be composed are equal or unequal: if they are equal, the 
cone would be equal to the circumscribed cylinder; but if unequal, 
they would be idented like steps. 36 We do not know how he resolved 
this aporia, but it has been suggested that he made use of the idea of 
infinitely thin circular laminae, or indivisibles, to find the volumes of 
cones and cylinders, anticipating and using Cavalieri's theorem for 
these special cases. 37 Democritus seems to have discriminated clearly 
br tween physical and mathematical atoms — as did his later follower 
Epicurus, although Aristotle made no such distinction 38 and according 
to the much later account of Simplicius Democritus is said to have 
held that all lines are divisible to infinity. 39 However, since most of 
Democritus' work is lost, we cannot now reconstruct his thought. 
That he was interested in other mathematical problems bearing on 
the infinitesimal we know from the titles of works now lost, but 
which are referred to by Diogenes Laertius. One of these seems to 
have been on horn angles (the angles formed by curves which have a 
common tangent at a point), and another on irrational (incommen- 
surable) lines and solids. 40 It may be inferred, therefore, that the 
Pythagorean difficulty with the incommensurable was probably 
familiar to him, and it may be that he tried to solve it by some theory 
of mathematical atomism. It has been maintained 41 that Democritus 
was too good a mathematician to have had anything to do with such 
a theory as that of indivisible lines; but it is difficult to imagine 
how a mathematical atom is to be conceived if not as an indivisible. 
At all events, whatever his conception of the nature of infinitesimals 
may have been, the influence of Democritus has persisted. The idea 
of the fixed infinitesimal magnitude has clung tenaciously to mathe- 
matics, frequently to be invoked by intuition when logic apparently 
failed to offer a solution, and finally to be displaced in the last century 
by the rigorous concepts of the derivative and the integral. 

35 Plutarch, Miscellanies and Essays, IV, 414-16. 

34 See also T. L. Heath, History of Greek Mathematics, I, 180; and Luria, op. cit., pp. 
138-40, for statements of this "paradox." 

" Simon, Geschichte der Mathematik im Altertum, p. 181. *• Luria, op. cit., pp. 179-80. 
M Cf. Simplicii commentarii in octo Aristotelis physicae auscultationis libros, p. 7. 
« T. L. Heath, History of Greek Mathematics, 1, 179-81. « Ibid., p. 181. 



Conceptions in Antiquity 23 

That the infinitesimal was not eagerly welcomed into Greek geom- 
etry after the time of the Pythagoreans and Democritus may have 
been due largely to a school of philosophy that had risen at Elea, in 
Magna Graecia. The Eleatic school, although not essentially mathe- 
matical, was apparently familiar with, and probably influenced by, 
Pythagorean mathematical philosophy; but it became an opponent of 
the chief tenet of this thought. Instead of proclaiming the constitution 
of objects as an aggregate of units, it pointed out the apparent con- 
tradictions inherent in such a doctrine, maintaining against the atomic 
view the essential oneness and changelessness of the world. This 
stultifying monism was upheld by Parmenides, the leader of the 
school, with perhaps a touch of skepticism derived from his iconoclastic 
predecessor, the poet-philosopher Xenophanes. 

In an indirect defense of this doctrine, the Eleatics proceeded to 
demolish, with skillful dialectic, the basis of opposing schools of 
thought. The most damaging arguments were offered by Zeno, the 
student of Parmenides. After presenting the obvious objection to the 
Pythagorean indefinitely small monad — that if it has any length, an 
infinite number will constitute a line of infinite length; and if it has 
no length, then an infinite number will likewise have no length — he 
added the following general dictum against infinitesimals: "That 
which, being added to another does not make it greater, and being 
taken away from another does not make it less, is nothing." 42 More 
critical and subtle than these, however, are his four famous paradoxes 
on motion. 43 There has been much speculation as to the purpose of 
Zeno's arguments, 44 lack of evidence making it impossible to decide 
conclusively against whom they were directed: whether against the 
Pythagoreans, or the atomists, or Heraclitus, or whether they were 
mere sophisms. That they were intended merely as dialectical puzzles 
may perhaps be indicated by the passage in Plutarch's life of Pericles: 

Also the two-edged tongue of mighty Zeno, who, 
Say whp.t one would, could argue it untrue. 45 

** Zeller, Die Philosopkie der Griechen in ihrer Geschichtlichen Entwicklung, I, 540. 

*• For an unusually extensive account of the history of Zeno's paradoxes, with bibli- 
ography, see Cajori's article on "History of Zeno's Arguments on Motion." 

** On this subject see Cajori, "The Purpose of Zeno's Arguments on Motion." This 
article includes an account of the varying interpretations, as well as extensive biblio- 
graphical notes. 

46 Plutarch, The Lives of the Noble Grecians and Romans, p. 185. 



24 Conceptions in Antiquity 

On the other hand there is some reason to suppose that these 
arguments were presented in connection with a more significant pur- 
pose. It is not improbable that Zeno, although he was neither a 
mathematician nor a physicist, propounded the paradoxes to point 
out the weakness in the Pythagorean definition of a point as unity 
having position, and in the resulting Pythagorean multiplicity which 
did not distinguish clearly betw r een the geometrical and the physical. 46 
Pythagorean science and mathematics had been concerned with 
form and structure, and not with flux and variability; but had the 
Pythagoreans applied their philosophy to the aspects of change in 
nature rather than to those of permanence, the resulting explanation 
of motion would have been in terms of concepts attacked by Zeno in 
his third and fourth paradoxes (those of the arrow and the stade), 47 in 
which space and time are assumed to be composed of indivisible 
elements. The arguments would hold equally well, of course, against 
mathematical atomism. The first two paradoxes (the dichotomy and 
the Achilles 46 ) are directed against the opposite conception, that of the 
infinite divisibility of space and time, and are based upon the impos- 
sibility of conceiving intuitively the limit of the sum of an infinite 
series. The four paradoxes are, of course, easily answered in terms of 

46 See Paul Tannery, La GeomUrie grecque, p. 124; cf . also the same author, "Le Concept 
scientifique du continu. Zenon d'Elee et Georg Cantor." Milhaud (Nouvelles ttudes, pp. 
153-54) and Cajori concur in the view here presented. See further, Cajori, "Purpose of 
Zeno's Arguments." 

47 The argument in the arrow is as follows: Anything occupying space equal to itself (or 
in one and the same place) is at rest; but this is true of the arrow at every moment of its 
flight. Therefore the arrow does not move. (See The Works of Aristotle, Vol. II, Physica VI. 
239b, for the statement of the paradoxes.) The argument in the stade, as given by Aristotle, 
is obscure (because of brevity), but is equivalent to the following: Space and time being 
assumed to be made up of points and instants, let there be given three parallel rows of 
points, A, B, and C. Let C move to the right and A to the left at the rate of one point per 
instant, both relative to B; but then each point of A will move past two points of C in an 
instant, so that we can subdivide this, the smallest interval of time; and this process can 
be continued ad infinitum, so that time can not be made up of instants. 

48 The argument in the dichotomy is as follows: before an object can traverse a given 
distance, it must first traverse half of this distance; before it can cover half, however, it 
must cover one quarter; and so ad infinitum. Therefore, since the regression is infinite, 
motion is impossible, inasmuch as the body would have to traverse an infinite number of 
divisions in a finite time. The argument in the Achilles is similar. Assume a tortoise to have 
started a given distance ahead of Achilles in a race. Then by the time Achilles has reached 
the starting point of the tortoise, the latter will have covered a certain distance; in the time 
required by Achilles to cover this additional distance, the tortoise will have gone a little 
farther; and so ad infinitum. Since this series of distances is infinite, Achilles can never 
overtake the tortoise, for the same reason as that adduced in the dichotomy. 



Conceptions in Antiquity 25 

the concepts of the differential calculus. There is no logical difficulty 
in the dichotomy or the Achilles, the uneasiness being due merely to 
failure of the imagination to realize, in terms of sense impressions, the 
nature of infinite convergent series which are fundamental in the 
precise explanation of, but not involved in our obscure notion of, 
continuity. The paradox of the flying arrow involves directly the 
conception of the derivative and is answered immediately in terms of 
this. The argument in this paradox, as also that in the stade, is met 
by the assumption that the distance and time intervals contain an 
infinite number of subdivisions. Mathematical analysis has shown that 
the conception of an infinite class is not self-contradictory, and that 
the difficulties here, as also in the case of the first two paradoxes, are 
those of conceiving intuitively the nature of the continuum and of 
infinite aggregates. 49 

In a broad sense there are no insoluble problems, but only those 
which, arising from a vague feeling, are not yet suitably expressed. 60 
This was the position of Zeno's paradoxes in Greek thought; for the 
notions involved were not given the precision of expression necessary 
for the resolution of the putative difficulties. It is clear that the 
answers to Zeno's paradoxes involve the notions of continuity, limits, 
and infinite aggregates — abstractions (all related to that of number) 
to which the Greeks had not risen and to which they were in fact 
destined never to rise, although we shall see Plato and Archimedes 
occasionally straining toward such views. That they did not do so 
may have been the result of their failure, indicated above in the case 
of the Pythagoreans, clearly to separate the worlds of sense and 
reason, of intuition and logic. Thus mathematics, instead of being the 
science of possible relations, was to them the study of situations 
thought to subsist in nature. 

The inability of Greek mathematicians to answer in a clear manner 
the paradoxes of Zeno made it necessary for them to forego the 
attempt to give to the phenomena of motion and variability a quanti- 
tative explanation. These experiences were consequently confined to 
the field of metaphysical speculation, as in the work of Heraclitus, or 

* Accounts of the mathematical resolutions of Zeno's paradoxes are given in the works 
of Bertrand Russell, The Principles of Mathematics and Our Knowledge of the External 
World. 

80 Enriques, Problems of Science, p. 5. 



36 Conceptions in Antiquity 

to that of qualitative description, as the physics of Aristotle. Only the 
static aspects of optics, mechanics, and astronomy found a place in 
Greek mathematics, and it remained for the Scholastics and early 
modern scientists to establish a quantitative dynamics. Zeno's argu- 
ments and the difficulty of incommensurability had also a more 
general effect on mathematics: in order to retain logical precision, it 
was necessary to give up the abortive Pythagorean effort to identify 
the domains of number and geometry, and to abandon also the pre- 
mature Democritean attempt to explain the continuous in terms of 
the discrete. It is, however, impossible satisfactorily to interpret the 
world of nature and the realm of geometry (spheres which for the 
Greeks were not essentially distinct) without superimposing upon 
them a framework of discrete multiplicity; without ordering, by means 
of number, the heterogeneity of impressions received by the senses; 
and without at every point comparing nonidentical elements. Thought 
itself is possible only in terms of a plurality of elements. As a conse- 
quence, the concept of discreteness cannot be excluded completely 
from the study of geometry. The continuous is to be interpreted in 
terms of successive subdivision, that is to say, in terms of the discrete, 
although from the Greek point of view the former could not be logically 
identified with the latter. The clever manner in which the method of 
successive subdivision was applied in Greek geometry, without the 
loss of logical rigor, will be seen later in the method of exhaustion — a 
procedure which was developed, not in Italy, but in and around the 
Greek mainland, whither many Pythagoreans wandered, on the 
breaking up of the school, toward the beginning of the fifth century 
b. c. Zeno likewise lived for a time in Athens, the rising center of 
Greek culture and mathematics. Here Pericles, the political leader 
of that city in its Golden Age, is said to have been one of his 
listeners. 81 

At Athens the great philosopher Plato, although himself not pri- 
marily a mathematician, was conversant with, and displayed a lively 
interest in, the problems of the geometers. He may not have contrib- 
uted much original work in mathematics, but he advanced the sub- 
ject, nevertheless, through his great enthusiasm for it. He is said to 
have paid particular attention to the principles of geometry — to the 

" Plutarch, Lives, p. 185. 



Conceptions in Antiquity 27 

hypotheses, definitions, methods. 62 For this reason he was particularly 
concerned with the difficulties which led eventually to the calculus. 
In his dialogues he considered the Pythagorean problem of the nature 
of number and its relationship to geometry, 53 the difficulty of incom- 
mensurability, 64 the paradoxes of Zeno, 66 and the Democritean ques- 
tion of indivisibles and the nature of the continuum. 56 

Plato seems to have realized the gulf between arithmetic and 
geometry, and it has been conjectured 67 that he may have tried to 
bridge it by his concept of number and by the establishment of arith- 
metic upon a firm axiomatic basis similar to that which was built up 
in the nineteenth century independently of geometry; but we cannot 
be sure, because these thoughts do not occur in his exoteric writings 
and were not advanced by his successors. If Plato made an attempt 
to arithmetize mathematics in this sense, he was the last of the 
ancients to do so, and the problem remained for modern mathe- 
matical analysis to solve. The thought of Aristotle we shall find 
diametrically opposed to any such conceptions. It has been suggested 
that Plato's thought was so opposed by the polemic of Aristotle that 
it was not even mentioned by Euclid. Certain it is that in Euclid 
there is no indication of such a view of the relation of arithmetic to 
geometry; but the evidence is insufficient to warrant the assertion 68 
that, in this connection, it was the authority of Aristotle which held 
back for two thousand years a transformation which the Academy 
sought to complete. A sound basis for either mechanics or arithmetic 
must be built upon the limit concept — a notion which is not found in 
the extant works of Plato nor in those of his successors. The Platonists, 
on the contrary, attempted to develop the misleading idea of indi- 
visibles or fixed infinitesimals, a notion which the modern arith- 
metization of analysis has had cause to reject. 

H To Plato are ascribed, among other things, the formulation of the analytic method 
and the restriction of Euclidean geometry to constructions possible with ruler and compass 
only. Hankel (Zur Gesckichte der Mathematik in Alterthum und Mittelalter, p. 156) ascribes 
this limitation to Plato, but T. L. Heath (History of Greek Mathematics, I, 288) would 
place it earlier. 

63 Republic VII. 525-27. 

M See, in particular, Theaetetus 147-48; Laws 819d-820c. 

66 Parmenides 128 ff. 
68 Philebas 17 ff. 

67 Toeplitz, "Das Verhaltnis von Mathematik und Ideenlehre bei Plato." 

68 Ibid., pp. 10-11. 



28 Conceptions in Antiquity 

Plato apparently did not give direct answers to the difficulties 
involved in incommensurability or in Zeno's paradoxes, although he 
expressed his opposition to the Pythagorean concepts of infinity, and 
of the monad as being unity having position, 69 and also to Democritean 
atomism. He was strongly influenced by both of these schools, but 
apparently felt that their views were too much the result of sense 
experience. Plato's criterion of reality was not consistency in experi- 
ence but reasonableness in thought. For him, as for the Pythagoreans, 
there was no necessary distinction between mathematics and science; 
both were the result of deduction from clearly perceived first principles. 
The Pythagorean monad and the Democritean mathematical atom- 
ism,, which gave every line a thickness, perhaps appealed too strongly 
to materialistic sense experience to suit Plato, so that he had recourse 
to the highly abstract apeiron or unbounded indeterminate. This was 
the eternally moving infinite of the Ionian philosopher, Anaximander, 60 
who had suggested it in opposition to Thales' less subtle assertion that 
the concrete material element, water, was the basis of all things. 
According to Plato, the continuum, could better be regarded as gen- 
erated by the flowing of the apeiron than thought of as consisting of 
an aggregation (however large) of indivisibles. This view represents a 
fusion of the continuous and the discrete not unlike the modern 
intuitionism of Brouwer. 61 The infinitely small was apparently not to 
be reached through a continued subdivision, 62 but was to be regarded, 
perhaps, as analogous to the generative infinitesimal of Leibniz, or 
the "intensive" infinitely small magnitude which appeared in ideal- 
istic philosophy in the nineteenth century. Mathematics has found it 
necessary to discard both views in making the infinitesimal subor- 
dinate to the derivative in the logical foundation of the calculus. 
However, the notion of the infinitesimal proved very suggestive in the 
early establishment of the calculus, and, as Newton remarked more 
than two thousand years later, the application to it of our intuitions 
of motion removes from the doctrine much of the harshness felt in 
the mathematical atomism of Democritus and later of Cavalieri. 
This, however, necessarily led to a loss both of precise logical defi- 

69 See T. L. Heath, History of Greek Matliematics, I, 293. 

60 Hoppe. "Zur Geschichte der Infinitesimalrechnung," p. 154; see also Milhaud, Les 
Philosophes giometres, p. 68. 

81 Helmholtz, Counting and Measuring, pp. xxii-xxiv. 82 Hoppe, op. cit., p. 152. 



Conceptions in Antiquity 29 

nition and of clear sensory interpretation, neither of which Plato 
supplied. 63 

The belief that mathematics becomes "sterilized by losing contact 
with the world's work" 64 is widely held but is not easily justified. 
The conjunction of mathematics and philosophy, as found, for ex- 
ample, in Plato, Descartes, and Leibniz, has been perhaps as valuable 
in suggesting new advances as has the blending of mathematical and 
scientific thought illustrated by Archimedes, Galileo, and Newton. 
The disregard in Platonic thought of any basis in the evidences of 
sense experience has not unjustly been regarded, from the scientific 
point of view, as an "unmitigated misfortune." On the other hand, 
the successful development of his views would have given to mathe- 
matics — which is interested solely in relationships which are logically 
thinkable rather than in those believed to be realized in nature — a 
flexibility and an independence of the world of sense impressions 
which were to be essential for the ultimate formulation of the con- 
cepts of the calculus. One may therefore say, in a very general sense, 
that "we know from Plato's own writings that he was thinking out 
the solution of problems that lead directly to the discovery of the 
calculus." 66 It is, however, altogether too much to assert that "Indeed 
there are probably only four or five names of mathematical dis- 
coverers that stand between Plato on the one hand and Newton and 
Leibniz, the discoverers of the calculus, on the other hand." 66 We 
shall see that the calculus was the result of a long train of mathe- 
matical thought, developed slowly and with great difficulty by very 
many thinkers. 

That the doctrines of the continuous and the infinitesimal did not 
develop along the abstract lines vaguely indicated by Plato was 
probably the result of the fact that Greek mathematics included no 
general concept of number, 67 and, consequently, no notion of a con- 
tinuous algebraic variable upon which such theories could logically 
have been based. Disregard of the abstract idealizations which Plato 

83 Hoppe {op. cit., p. 152) asserts that in Plato one finds the first clear conception of the 
infinitesimal. It is difficult, however, to perceive on what grounds such a thesis is to be 
defended. 

84 Hogben, Science for the Citizen, p. 64. 

66 Marvin, The History of European Philosophy, p. 142. 
M Ibid. 

67 Miller, "Mathematical Weakness of Early Civilizations." 



S6 Conceptions in Antiquity- 

suggested, but never clearly defined, may also have been due in some 
measure to the opposition afforded by the inductive scientific views of 
Aristotle and the Peripatetic School. Aristotelian thought, while not 
destroying the rigorously deductive character of Greek geometry, may 
have preserved in Greek mathematics that strong reasonable and 
matter-of-fact cast which one finds in Euclid and which operated 
against the early development of the calculus, as well as against the 
Platonic tendency toward speculative metaphysics. 

Although Plato did not solve the difficulties which the Pythagoreans 
and Democritus encountered, he urged their study upon his asso- 
ciates, inveighing against the ignorance concerning such problems 
which prevailed among the Greeks. 68 To Eudoxus he is said to have 
proposed a number of problems in stereometry which proved to be 
remarkably suggestive in leading toward the calculus. In this con- 
nection the demonstrations which Eudoxus gave of the propositions 
(previously stated without proof by Democritus) on the volumes of 
pyramids and cones led to his famous general method of exhaustion 
and to his definition of proportion. The achievements of Eudoxus are 
those of a mathematician who was at the same time a scientist, with 
none of the occult or mystic in him. 69 As a consequence, they are based 
at every point on finite, intuitively clear, and logically precise consid- 
erations. In method and spirit the later work of Euclid will be found 
to owe much more to Eudoxus than to Plato. 

We have seen that the Pythagorean theory of proportion could not 
be applied to all lines, many of which are incommensurable, and 
that the Democritean view of infinitesimals was logically untenable. 
Eudoxus proposed means by which these difficulties could be avoided. 
The paths he indicated, in his theory of proportion and in the method 
of exhaustion, were not the equivalents of our modern conceptions of 
number and limit, but rather detours which obviated the necessity of 
using the latter. They were significant, however, in that they made it 
possible for the Greek mind confidently to pursue its attack upon prob- 
lems which were to eventuate much later in the calculus. 

The Pythagorean conception of proportion had been the result of 

the identification of geometrical magnitudes and integral numbers. 

M See Laws 819d-820c. 

•» T. L. Heath, History of Greek Mathematics, I, 323-25; Becker, "Eudoxos-Studien," 
1936, p. 410. 



Conceptions in Antiquity 31 

Two lines, for example, were to each other as the ratio of the (integral) 
numbers of units in each. With the discovery of the incommensura- 
bility of some lines with others, however, this definition could no longer 
be universally applied. Eudoxus substituted for it another which was 
more general, in that it did not require two of the terms in the propor- 
tion to be (integral) numbers, but allowed all four to be geometrical 
entities, and required no extension of the Pythagorean idea of number. 
Euclid 70 states Eudoxus' definition as follows: "Magnitudes are said 
to be in the same ratio, the first to the second and the third to the 
fourth, when, if any equimultiples whatever be taken of the first 
and third, and any equimultiples whatever of the second and fourth, 
the former equimultiples alike exceed, are alike equal to, or alike fall 
short of, the latter equimultiples respectively taken in corresponding 
order." 71 The theory of proportion thus stated involves only geo- 
metrical quantities and integral multiples of them, so that no general 
definition of number, rational or irrational, is necessary. It is interesting 
to see that after the development of mathematical analysis, the con- 
cept of proportion resembles the arithmetical form of the Pythag- 
oreans rather than the geometrical one of Eudoxus. Even when the 
ratio is not expressible as the quotient of two integers, we now sub- 
stitute for it a single number and symbol such as v or e. Although 
Eudoxus did not, as we do, regard the ratio of two incommensurable 
quantities as a number, 72 nevertheless his definition of proportion ex- 
presses the ordinal idea involved in the present conception of real 
number. The assertion that it is "word for word the same as the 
general definition of number given by Weierstrass" 73 will be found, 
however, to be incorrect, both literally and in its implications. The 
formulation of Eudoxus was, on the contrary, a means of avoiding the 
need of such an arithmetic definition as that of Weierstrass. 

The method of exhaustion of Eudoxus shows the same abandon- 
ment of numerical conceptions which we have seen in his theory of 
proportion. Length, area, and volume are now carefully defined 
numerical entities in mathematics. After the time of the Pythagoreans, 
classic Greek mathematics did not attempt to identify number with 

70 Book V, Definition 5. 

n The Thirteen Books of Euclid's Elements, trans, by T. L. Heath, II, 114. 
n Stolz, Vorlesungen iiber aUgemeine Arithmetik, I, 94. 

71 Simon, "Historische Bemerkungen iiber das Continuum," p. 387. 



32 Conceptions in Antiquity 

geometrical quantities. As a result no rigorous general definitions of 
length, area, and volume could then be given, the meaning of these 
quantities being tacitly understood as known from intuition. The 
question, "What is the area of a circle?" would have had no meaning 
to the Greek geometers. But the query, "What is the ratio of the 
areas of two circles?" would have been a legitimate one, and the 
answer would have been expressed geometrically: "the same as that 
of squares constructed on the diameters of the circles." 74 The fact that 
squares and circles are incommensurable with each other does not 
cause any incongruity in the idea of their entering into the same pro- 
portion under the general definition of Eudoxus; but the proof of the 
correctness of the proportion requires in this case the comparison of 
squares with squares and of circles with circles. 

Obviously the old Pythagorean method of the application of areas 
cannot be employed in the case of circles, so Eudoxus had recourse 
to an idea which had been advanced sometime before by Antiphon 
the Sophist and again a generation later by Bryson. These men had 
inscribed within a circle a regular polygon, and by successively 
doubling the number of sides they seem to have hoped to reach a 
polygon which would coincide with the circle and so "exhaust" its 
area. It should, however, be borne in mind that we do not know just 
what Antiphon (and later Bryson) said. The method of Antiphon has 
been described 75 as equivalent at one and the same time to the method 
of Eudoxus (as given in Euclid XII, 2), and to our conception of the 
circle as the limit of such an inscribed polygon, but merely expressed 
in different terminology. This cannot be strictly correct. If Antiphon 
had considered the process of bisection as carried out to an infinite 
number of steps, he would not have been thinking in the terms of 
Eudoxus and Euclid, as we shall see. If, on the other hand, he did not 
regard the process as continued indefinitely but only as carried out to 
any desired degree of approximation, he could not have had our idea 
of a limit. Furthermore, our conception of the limit is numerical, 
whereas the notions of Antiphon and Eudoxus are purely geometrical. 

The suggestive idea of Antiphon, however, was adopted by Bryson, 
who is reputed not only to have inscribed a polygon within the circle 

74 Cf. Vogt, Der Grenzbegrijf in der Elementar-mathematik, p. 42. 
78 T. L. Heath, History of Greek Mathematics, I, 222. 



Conceptions in Antiquity 33 

but also to have circumscribed one about it as well, saying that the 
circle would ultimately, as the result of continued bisection, be the 
mean of the inscribed and circumscribed polygons. Again we do not 
know exactly what he said, and cannot tell clearly what he meant. 76 
Interpretations have been advanced 77 which would go so far as to see 
in the work of Bryson the concept of a "Dedekind Cut" or of the con- 
tinuum of Georg Cantor, but the evidence would hardly appear to 
warrant such imputations. However, the idea which he suggested was 
developed by Eudoxus into a rigorous tyDe of argument for dealing 
with problems involving two dissimilar, heterogeneous, or incom- 
mensurable quantities, in which intuition fails to represent clearly the 
transition from one to the other which is necessary to make a com- 
parison possible. 

The procedure which Eudoxus proposed has since become known 
as the method of exhaustion. The principle upon which this method 
is based is commonly called the lemma, or postulate, of Archimedes, 
although the great Syracusan mathematician himself ascribed it 78 to 
Eudoxus and it is not improbable that it had been formulated still 
earlier by Hippocrates of Chios. 79 This axiom (as given in Euclid X, 
1) states that, given two unequal magnitudes (neither equal to zero, 
of course, since for the Greeks this was neither a number nor a mag- 
nitude), "if from the greater there be subtracted a magnitude greater 
than its half, and from that which is left a magnitude greater its half, 
and if this process be repeated continually, there will be left some 
magnitude which will be less than the lesser magnitude set out." 80 
This definition (in which, of course, any ratio may be substituted in 
place of one-half) excluded the infinitesimal from all demonstrations 
in the geometry of the Greeks, although we shall find this banished 
notion entering occasionally into their thought as an explorative aid. 
From the fact that, on continuing the process indicated in the axiom 
of Archimedes, the magnitude remaining can be made as small as we 
please, the procedure introduced by Eudoxus came much later to be 

74 Ibid., I, 224. 

77 See Becker, "Eudoxos-Studien," in particular, 1933, pp. 373-74; Toeplitz, "Das Ver- 
haltnis von Mathematik und Ideenlehre bei Plato," pp. 31-33. 

78 In his Quadrature of the Parabola. See T. L. Heath, History of Greek Mathematics, I, 
327-28. 

79 Hankel, Zur Geschichte der Mathematik in Alterthum und Mittelalter, p. 122. 

80 Euclid, Elements, Heath trans., vol. HI, p. 14. 



34 Conceptions in Antiquity 

called the method of exhaustion. It is to be remarked, however, that 
the word exhaustion was not applied in this connection until the 
seventeenth century, 81 when mathematicians somewhat ambiguously 
and uncritically employed the term indifferently to designate both the 
ancient Greek procedure and their own newer methods which led 
immediately to the calculus and which truly "exhausted" the mag- 
nitudes. 

The Greek mathematicians, however, never considered the process 
as being literally carried out to an infinite number of steps, as we do 
in passing to the limit — a concept which allows us to interpret the 
area or volume as truly exhausted, or at least as defined as the limit 
of the infinite numerical sequence obtained in this manner. There 
was always, in the Greek mind, a quantity left over (although this 
could be made as small as desired), so that the process never passed 
beyond clear intuitional comprehension. A simple illustration will per- 
haps serve to make the nature of the method clear. The proposition, 
in Euclid XII, 2, that the areas of circles are to each other as the 
squares on their diameters will suffice for this purpose. The substance 
of this proof is as follows: Let the areas of the circles be A and a, and 
let their diameters be D and d respectively. If the proportion a : A 
= d 2 :D 2 is not true, then let a' : A = d 2 : D 2 , where a' is the area 
of another circle either greater or smaller than a. If a' is smaller 
than a, then in the circle of area a we can inscribe a polygon of area p 
such that p is greater than a' and smaller than a. This follows from 
the principle of exhaustion (Euclid X, 1) — that if from a magnitude 
(such as the difference in area between a' and a) we take more than 
its half, and from the difference more than its half, and so on, the 
difference can be made less than any assignable magnitude. If P is 
the area of a similar polygon inscribed in the circle of area A, then 
we know that p : P — d 2 : D 2 = a' :A. But since p > a', then P > 
A, which is absurd, since the polygon is inscribed within the circle. 
In a similar manner it can be shown that the supposition a! > a like- 
wise leads to a contradiction, and the truth of the proposition is 
therefore established. 82 

The method of exhaustion, although equivalent in many respects 

81 In particular in Gregory of St. Vincent, Opus geometricum, pp. 739-40, 

82 Cf. Euclid, Elements, Heath trans., vol. Ill, pp. 371-78. 



Conceptions in Antiquity 35 

to the type of argument now employed in proving the existence of a 
limit in the differential and the integral calculus, does not represent 
the point of view involved in the passage to the limit. The Greek method 
of exhaustion, dealing as it did with continuous magnitude, was 
wholly geometrical, for there was at the time no knowledge of an 
arithmetical continuum. This being the case, it was of necessity based 
on notions of the continuity of space — intuitions which denied any 
ultimate indivisible portion of space, or any limit to the divisibility 
in thought of any line segment. The inscribed polygon could be made 
to approach the circle as nearly as desired, but it could never become 
the circle, for this would imply an end in the process of subdividing 
the sides. However, under the method of exhaustion it was not neces- 
sary that the two should ever coincide. By an argument based upon 
the reductio ad absurdum, it could be shown that a ratio greater or 
less than that of equality was inconsistent with the principle that the 
difference could be made as small as desired. 

The argument of Eudoxus appealed at every stage to intuitions of 
space, and the process of subdivision made no use of such unclear con- 
ceptions as that of a polygon with an infinite number of sides — that 
is, of a polygon which should ultimately coincide with the circle. 
No new concepts were involved, and the gap between the curvilinear 
and the rectilinear still remained unspanned by intuition. Eudoxus, 
however, had most ingeniously contrived to demonstrate — without 
resort to the logically self -contradictory infinitesimal previously 
invoked by vague imagination — the truth of certain geometrical 
propositions requiring a comparison of the curvilinear with the 
rectilinear and of the irrational with the rational. 

There is no logical difficulty to be found in the argument used in 
the method of exhaustion, but the cumbersomeness of its application 
led later mathematicians to seek a more direct approach to problems 
in which the application of some such procedure would have been 
indicated. The method of exhaustion has, most misleadingly, been 
characterized as "a well-established algorithm of the differential cal- 
culus." 83 It is indeed true that the problems to which the method was 
applied were those which led toward the calculus. Nevertheless, it is 
not incorrect to say that the procedure involved actually directed 

83 Simon, "Zur Geschichte und Philosophic der Differentialrechnung," p. 116. 



36 Conceptions in Antiquity- 

attention away from the discovery of an equivalent algorithm, in that 
it directed attention toward the synthetic form of exposition rather 
than toward an analytic instrument of discovery. 84 It did represent 
a conventional type of demonstration, but the Greek mathematicians 
never developed this into a concise and well-recognized operation with 
a characteristic notation. In fact the ancients never made the first 
step in this direction: they did not formulate the principle of the 
method as a general proposition, reference to which might serve in 
lieu of the argument by the ubiquitous double reductio ad absurdum. 85 

It was largely in connection with the search for some means of 
simplifying the arguments in the tedious methods of the ancients that 
the differential and integral calculus was developed in the seventeenth 
century. To trace this development is the purpose of this essay; but 
at this point it may not be amiss to anticipate the final formulation 
to the extent of comparing the nature of the basic concept of the 
calculus — that of the limit of an infinite sequence — with the view 
indicated in the method of exhaustion. 

The limit of the infinite sequence P h P 2 , . . . , P w , . . . (the 
terms of which represent, for example, the areas of the inscribed 
polygons considered in the proposition above) has been defined in the 
introduction to be the number C, such that, given any positive num- 
ber e, we can find a positive integer N, such that f or n > N it can 
be shown that \ C — P n \ < e. The spatial intuition of the method 
of exhaustion, with its application of areas, unlimited subdivision, 
and argumentation by a reductio ad absurdum, here gives way to 
definition in terms of formal logic and number, i. e., of infinite ordered 
aggregates of the positive integers. The method of exhaustion corre- 
sponds to an intuitional concept, described in terms of mental pic- 
tures of the world of sensory perception. The notion of a limit, on 
the other hand, may be regarded as a verbal concept, the explication 
of which is given in terms of words and symbols — such as number, 
infinite sequence, less than, greater than — with regard not to any 
mental visualization, but only to their definition in terms of the 
primary undefined elements. The limit concept is thus by no means 
to be considered ineffable; nor does it imply that there is other than 

84 Cf. Brunschvicg, Les tiapes de la philosophic mathtonatique, pp. 157-59. 
86 The Works of Archimedes, ed. by T. L. Heath, p. cxliii. 



Conceptions in Antiquity 37 

empirical experience. It simply makes no appeal to intuition or 
sensory perception. It resembles the method of exhaustion in that it 
allows our vague instinctive feeling for continuity to shift for itself 
in any effort that may be made to picture how the gap between the 
curvilinear and the rectilinear, or between the rational and the irra- 
tional, is bridged, for such an attempt is quite irrelevant to the logical 
reasoning involved. The limit C is not for this reason to be regarded 
as a sophistic or inconceivable quantity which somehow nevertheless 
enters into real relations with other similar quantities, nor is it to 
be visualized as the last term of the infinite sequence. It is to be 
considered merely as a number possessing the property stated in the 
definition. It is to be borne in mind that although adumbrations of the 
limit idea appear in the history of mathematics in ancient times, 
nevertheless the rigorous formulation of this concept does not appear 
in work before the nineteenth century — and certainly not in the 
Greek method of exhaustion. 86 

The apparent break, in the mathematical work of Eudoxus, from 
the metaphysical aspect of Platonism is seen equally clearly 87 in the 
philosophical thought of one who studied under Plato for twenty 
years and who was known as the "mind of the School." 88 Aristotle 
borrowed freely from the work of his predecessors, with the result 
that, although not primarily a mathematician, he was familiar with 
the difficulties and results of Greek mathematics, including the method 
of exhaustion. He wrote a work (now lost) On the Pythagoreans, dis- 
cussed at some length the paradoxes of Zeno, mentioned Democritus 
frequently in mathematics and science (although always to refute 
him), was intimately familiar with Plato's thought, and was ac- 
quainted with the work of Eudoxus. In spite of his competence in 
mathematics and of his frequent use of geometry in his constructions, 89 
Aristotle's approach to the problems involved was essentially scientific, 
in the inductively descriptive sense. Furthermore, for Plato's mathe- 
matical intellectualism he substituted a grammatical intuitionism. 90 

86 Milhaud {Les Pkttosophes glomitres, p. 182) would have Eudoxus consider the circles 
as the limit of a polygon, but this could not have been in the sense in which the term limit 
is now employed. 

87 Cf., in this respect, Jaeger, Aristoteles. w Ross, Aristotle, p. 2. 

89 Enriques, Problems of Science, p. 110. See also Gorland, Aristoteles und die Mathe- 
matik, and Heiberg, "Mathematisches zu Aristoteles." 

90 Brunschvicg, Les Stapes de la philosophic mathimatique, p. 70. 



38 Conceptions in Antiquity 

Although he realized that the objects of mathematics are not those 
of sense experience and that the figures used in demonstrations are for 
illustration only 91 and in no way enter into the reasoning, Aristotle's 
whole attitude was governed by a strong dependence upon the evi- 
dence of the senses, as well as upon logic, and by an aversion to 
abstraction and extrapolation beyond the powers of sensory percep- 
tion. As a consequence, he did not think of a geometrical line, as had 
Plato, as an idea which is prior to, and independent of, experience of 
the concrete. Neither did he regard it, as does modern mathematics, 
as an abstraction which is suggested, perhaps, though not in any 
way defined, by physical objects. He viewed it rather as a character- 
istic of natural objects which has merely been separated from its 
irrelevant context in the world of nature. "Geometry investigates 
physical lines but not qua physical," he said, 92 and added: 

Necessity in mathematics is in a way similar to necessity in things which 
come to be through the operation of nature. Since a straight line is what 
it is, it is necessary that the angles of a triangle should equal two right 
angles. 93 

The decisions which Aristotle rendered on the indivisible, the infinite, 
and the continuous were consequently those dictated by common 
sense. In fact, with the exception of Plato's successors in the Academy 
and, perhaps, of Archimedes, they were those accepted by the body of 
Greek mathematicians after Eudoxus. 

Only in the case of the indivisible, however, do Aristotle's views 
coincide with the present notions in mathematics. Modern science has 
opposed, modern mathematics upheld, Aristotle in his vigorous denial 
of the indivisible, physical and mathematical, of the atomic school. 
Recent physical and chemical theories of the atom have furnished a 
description of natural phenomena which offers a higher degree of 
consistency within itself and with sensory impressions than had the 
Peripatetic doctrine of continuous substantiality. Science has conse- 
quently, under Carneades' doctrine of truth, accepted the atom as a 
physical reality. Modern mathematics, on the other hand, agrees with 
Aristotle in his opposition to minimal indivisible line segments; not, 
however, because of any argument from experience, but because it has 

91 T. L. Heath, History of Greek Mathematics, I, 337. 

« Physica II. 193b-194a. » Physica TL. 200a. 



Conceptions in Antiquity 39 

been unable to give a satisfactory definition and logical elaboration of 
the concept. However, the mathematical indivisible, in spite of the 
opposition of Aristotle's authority, was destined to play an important 
part in the development of the calculus, which in the end definitely 
excluded it. That the concept enjoyed an extensive popularity even in 
Aristotle's day, Greek logic notwithstanding, is seen by the fact that 
a Peripatetic treatise formerly ascribed to Aristotle (but now thought 
to have been written by Theophrastus, or Strato of Lampsacus, or 
perhaps by someone else), the De lineis insecabilibus, u was directed 
against it. This presents many arguments against the assumption of 
indivisible lines and concludes that "it conflicts with practically every- 
thing in mathematics." 95 

The work may have been composed as an answer to Xenocrates, 
the successor of Plato in the Academy, who apparently maintained 
the existence of mathematical indivisibles. It has been asserted 96 that 
neither Aristotle nor Plato's successors understood the infinitesimal 
concept of their master, and that only Archimedes rose to a correct 
appreciation of it. It is to be remarked, however, that such an assump- 
tion is wholly gratuitous. Plato in his extant works offered no clear 
definition of the infinitesimal. Archimedes, moreover, made no men- 
tion of any indebtedness to Plato in this matter, and, as will be-seen, 
explicitly disclaimed any intention of regarding infinitesimal methods 
as constituting valid mathematical demonstrations. 97 The opposition 
of Aristotle to the doctrine of infinitesimals was wholly justified by 
considerations of logic, although from the point of view of the subse- 
quent development of the calculus the uncritical use of the infinitely 
small was for a time most fruitful. 

As in the case of the infinitesimal, so also with respect to the infinite 

the views of Aristotle constitute excellent illustrations of his abiding 

confidence in the ultimate interpretability of phenomena in terms of 

distinctly clear concepts derived from sensory experience. 98 The 

Pythagoreans had regarded space as infinitely divided, and Democritus 

had likewise spoken of the atoms as infinite in number. Plato, influ- 

M The Works of Aristotle, Vol. VI, Opuscula. K De lineis insecabilibtts 970a. 

94 Hoppe, "Zur Geschichte der Infinitesimalrechnung," p. 152. 
91 The Method of Archimedes, ed. by T. L. Heath, p. 17. 

w Aristotle's view of the infinite has been much discussed. For one of the most recent 
philosophical discussions of this subject, see Edel, Aristotle's Theory of the Infinite. 



40 Conceptions in Antiquity 

enced by these views and perhaps also by those of Anaximander and 
Anaxagoras, had not clearly distinguished the concrete from the 
abstract." He had held that the infinite was located at the same time 
in ideas and in the sensible world, 100 the line as made up of points being 
one illustration of this fact. However, no one of Aristotle's predecessors 
had made quite clear his position with respect to the infinite. Anaxag- 
oras, at least, seems to have realized that it is only the imagination 
which objects to the infinite and to an infinite subdivision. 101 

On the other hand, Aristotle, in adopting the inductive scientific 
attitude, did not go beyond what is clearly representable in the mind. 
In consequence he denied altogether the existence of the actual infinite 
and restricted the use of the term to indicate a potentiality only. 102 
His clear distinction between an existent infinity and a potential 
infinity was the basis of much of the discussion of the Scholastics on 
the subject and of later controversies on the metaphysics of the 
calculus. His refusal to recognize the actual infinite was in keeping 
with his fundamental tenet that the unknowable exists only as a 
potentiality: that anything beyond the power of comprehension is 
beyond the realm of reality. Such a methodological definition of 
existence has led investigators in inductive science to continue to the 
present time the Aristotelian attitude of negation toward the infi- 
nite; 103 but such a view, if adopted in mathematics, would exclude 
the concepts of the derivative and the integral as extrapolations 
beyond the thinkable, and would, in fact, reduce mathematical 
thought to the intuitively reasonable. 

That the Aristotelian doctrine of the infinite was abandoned in the 
mathematics of the nineteenth century was largely the result of a 
shift of emphasis from the infinite of geometry to that of arithmetic; 
for in the latter field assumptions appear to be less frequently dictated 
by experience. For Aristotle such a change of view would have been 
impossible, inasmuch as his conception of number was that of the 
Pythagoreans: a collection of units. 104 Zero was not included, of 
course; nor was "the generator of numbers," the integer one. "The 

99 Paul Tannery, Pour Vhistoire de la science Hellene, pp. 300-5. 

100 Brunschvicg, Les Etapes de la philosophic mathimatique, p. 67. 

101 Paul Tannery, Pour Vhistoire de la science Hellene, pp. 293-94. 

102 Physica III. 206b. ** Barry, The Scientific Habit of Thought, p. 197. 
104 Physica HI. 207b; cf. also Plato, Republic VII. 525e. 



Conceptions in Antiquity 41 

smallest number in the strict sense of the word 'number' is two," 
said Aristotle. 106 Such a view of number could not be reconciled with 
the infinite divisibility of continuous magnitude which Aristotle 
upheld so vigorously. When, then, Aristotle distinguished two kinds of 
(potential) infinite — one in the direction of successive addition, or the 
infinitely large, and the other in the direction of successive subdivision, 
or the infinitely small — we find the behavior of number to be quite 
different from that of magnitude: 

Every assigned magnitude is surpassed in the direction of smallness, while 
in the other direction there is no infinite magnitude .... Number on the 
other hand is a plurality of "ones" and a certain quantity of them. Hence 
number must stop at the indivisible. . . . But in the direction of largeness 
it is always possible to think of a larger number. . . . Hence this infinite is 
potential, .... and not a permanent actuality but consists in a process 
of coming to be, like time .... With magnitudes the contrary holds. 
What is continuous is divided ad infinitum, but there is no infinite in the 
direction of increase. For the size which it can potentially be, it can also 
actually be. 106 

In commenting on the view of mathematicians, Aristotle said, 

In point of fact they do not need the infinite and do not use it. They pos- 
tulate only that the finite straight line may be produced as far as they 
wish. . . . Hence, for the purposes of proof, it will make no difference to 
them to have such an infinite instead, while its existence will be in the 
sphere of real magnitude. 107 

How well this characterizes Greek geometry can be seen in the method 
of exhaustion as presented by Eudoxus slightly earlier than Aristotle 
and by Euclid a little later. This method assumes in the proof only 
that the bisection can be continued as far as one may wish, not car- 
ried out to infinity. How far it lies from the point of view of modern 
analysis is indicated by the fact that the latter has been called "the 
symphony of the infinite." 

That Aristotle would be unable to cope with the problems of the 
continuous is perhaps to be expected, both from his view of the 
infinite and from the lack among the Greeks of an adequate arith- 
metical point of view. After having considered place and time in the 
fourth book of the Physica, and change in the fifth, Aristotle turned 
in the sixth book to the continuous. His account is based upon a 

105 Physica IV. 220a. m Physica in. 207b. 1W Ibid. 



42 Conceptions in Antiquity 

definition derived from the intuitive notion of the essence of con- 
tinuous magnitude: "By continuous I mean that which is divisible 
into divisibles that are infinitely divisible." 108 This view was sup- 
plemented by a naive appeal to the instinctive feeling of the necessity 
of a hang-togetherness — of the coincidence of the extremities of the 
component parts. 109 For this reason Aristotle denied that number can 
produce a continuum, 110 inasmuch as there is no contact in numbers. 111 
Only a generation ago this Aristotelian view had not been entirely 
abandoned, 112 but the mathematical continuum accepted at the present 
time is defined precisely in terms of the concept of number, or of 
classes of elements, and of the notion of separation — as in the Dede- 
kind Cut — rather than contact. The nature of continuous magnitude 
has been found to lie deeper than Aristotle believed, and it has been 
explained on the basis of concepts which require a broader definition 
of number than that held during the Greek period. The Aristotelian 
dictums on the subject were not unfruitful, however, for they led to 
speculations during the medieval period which in turn aided in the 
rise of the calculus and the modern doctrine of the continuum. 

In connection with the study of continuous magnitude, Aristotle 
attempted also to clarify the nature of motion, criticizing the atomists 
for their neglect of the whence and the how of movement. 113 Although 
he was quite adept at detecting problems, he failed to make his 
formulation of these quantitative and was consequently infelicitous in 
their resolution. In the light of modern scientific method, this lack of 
mathematical expression gives to his treatment of motion and varia- 
bility the appearance of a dialectical exercise, rather than of a serious 
effort to establish a sound basis for the science of dynamics. 114 

Aristotle's approach to the subject was qualitative and metaphys- 
ical. This is evidenced by his definition of motion as "the fulfillment 
of what exists potentially, in so far as it exists potentially," and by 
the further remark, "We can define motion as the fulfillment of the 
movable qua movable." 115 We shall find this qualitative explanation 
of motion — the result of the striving of a body to become actually 
what it is potentially — involved in less teleological forms in the idea 

108 Physica VI. 232b. 1M Physica VI. 231a; cf. also Categories 5a. 

110 Metaphysica 1075b and 1020a; Categoriae 4b. m Metaphysica 1085a. 

112 See Mere, A History of European Thought in the Nineteenth Century, II, 644. 
m Metaphysica 985b and 1071b. 
114 Cf. Mach, The Science of Mechanics, p. 511. m Physica III, 201a-202a. 



Conceptions in Antiquity 43 

of impetus developed by the Scholastics, in Hobbes' explanation of 
velocity and acceleration in terms of a conatus, and even in meta- 
physical interpretations of the infinitesimal of the calculus as an 
intensive quantity — that is, as a "becoming" rather than a "being." 
In this respect Aristotle's work may have encouraged the elaboration 
of notions leading toward the derivative. However, his influence was 
in another sense quite adverse to the development of this concept in 
that it centered attention upon the qualitative description of the change 
itself, rather than upon a quantitative interpretation of the vague 
instinctive feeling of a continuous state of change invoked by Zeno. 
The calculus has shown that the concept of continuous change is no 
more free from that of the discrete than is the numerical continuum, 
and that it is logically to be based upon the latter, as is also the idea 
of geometrical magnitude. As long as Aristotle and the Greeks con- 
sidered motion continuous and number discontinuous, a rigorous 
mathematical analysis and a satisfactory science of dynamics were 
difficult of achievement. 

The treatment of the infinite and of continuous magnitude found 
in the Physica of Aristotle has been regarded as presenting the ap- 
pearance of a veritable introduction to a treatise on the differential 
calculus. 116 Such a view, however, is seen to be most unwarranted, 
inasmuch as Aristotle expressed his unqualified opposition to the funda- 
mental idea of the calculus — that of an instantaneous rate of change. 
He asserted that "Nothing can be in motion in a present. . . . Nor 
can anything be at rest in a present." 117 This point of view neces- 
sarily operated against the mathematical representation of the phe- 
nomena of change and against the development of the calculus. 
Aristotle's denial of instantaneous velocity, as realized in the world 
described by science, is, to be sure, in conformity with the recognized 

limitations of sensory perception. Only average velocities, — , are 

At 

recognizable in this sense. In the world of thought, on the other hand, 
it has been found possible — through the calculus and the limit con- 
cept — to give a rigorous quantitative definition of instantaneous 

. ds 
velocity, — . Aristotle, however, in conformity with a view widely ac- 

CLt 

»• Moritz Cantor, "Origines du calcul infinit&imal," p. 6. u7 Physica VL 234a. 



44 Conceptions in Antiquity 

cepted at the time, regarded mathematics as a pattern of the world 
known through the senses and consequently did not foresee such a 
possibility. 

The failure of Aristotle to distinguish sharply between the worlds of 
experience and of mathematical thought resulted in his lack of clear 
recognition of a similar confusion in the paradoxes of Zeno. Aristotle 
refuted the arguments in the stade and the arrow by an appeal to 
sensory perception and the denial of an instantaneous velocity. 
Modern mathematics, on the other hand, has answered them in terms 
of thought alone, based on the concept of the derivative. In the same 
manner Aristotle resolved the paradoxes in the dichotomy and the 
Achilles by the curt assertion, suggested by experience, that although 
one cannot traverse an infinite space in finite time, it is possible to 
cover an infinitely divided space in finite time because of the infinite 
divisibility of the latter. 118 

Mathematics has, of course, given the solution of the difficulties in 
terms of the abstract concept of converging infinite series. In a certain 
metaphysical sense this notion of convergence does not answer Zeno's 
argument, in that it does not tell how one is to picture an infinite 
number of magnitudes as together making up only a finite magnitude; 
that is, it does not give an intuitively clear and satisfying picture, in 
terms of sense experience, of the relation subsisting between the_ 
infinite series and the limit of this series. If one demands that Zeno's 
paradoxes be answered in terms of our vague instinctive feeling for 
continuity — as essentially different from the discrete — no answers 
more satisfying than those of Aristotle (to whom we owe also the 
statement of the paradoxes, since we do not have Zeno's words) 
have been given. The unambiguous demonstration that the difficulties 
implied by the paradoxes are simply those of visualization and not 
those of logic was to require more precise and adequate definitions 
than any which Aristotle could furnish for such subtle notions as 
those of continuity, the infinite, and instantaneous velocity. Such 
definitions were to be given in the nineteenth century in terms of the 
concepts of the calculus; and modern analysis has, upon the basis of 
these, clearly dissented from the Aristotelian pronouncements in this 
field. The views of Aristotle are not on this account to be regarded — 

™ Physica VI. 239b-240a. 



Conceptions in Antiquity 45 

as is all too frequently and uncritically maintained 119 — as gross mis- 
conceptions which for two thousand years retarded the advancement 
of science and mathematics. They were, rather, matured judgments 
on the subject which furnished a satisfactory working basis for later 
investigations which were to result in the science of dynamics and in 
the mathematical continuum. Nevertheless, there is apparent in the 
work of Aristotle the cardinal weakness of Greek logic and geometry: 
a naive realism which regarded thought as a true copy of the external 
world. 120 This caused him to place too ingenuous a confidence in cer- 
tain instinctive feelings with respect to continuous magnitude and to 
seek, of all possible representations, that which presented the greatest 
plausibility in the light of sensory experience, rather than that which 
offered the widest consistency in thought. 

It has been said 121 that the fifth book of Euclid's Elements and the 
logic of Aristotle are the two most unobjectionable and unassailable 
treatises ever written. The two men were roughly contemporaries: 
Aristotle lived from 384 to 322 b. c. ; Euclid's birth has been placed at 
about 365 b. c. and the composition of the Elements may be accepted 
as between 330 and 320 b. c. 122 There is also a marked similarity 
between the Aristotelian apodictic and the mathematical method 
built up by Euclid. 123 Although Euclid was probably taught by the 
pupils of Plato, 124 the influence of the sober, hard-headed, scientific 
thought illustrated by Eudoxus and Aristotle must have predominated 
over the more abstract, speculative, and even mystical trend seen in 
the immediate successors of Plato to the leadership of the Academy 
and carried to excess by later Neoplatonists. There is in Euclid none 
of the metamathematics which played such a prominent part in 
Plato's thought, nor do metaphysical speculations on mathematical 
atomism enter. Mathematics was regarded by Euclid neither as a 
necessary form of cosmological intelligibility, nor as a mere tool of 

119 See, for example, Mayer, "Why the Social Sciences Lag behind the Physical and Bio- 
logical Sciences"; cf. also Toeplitz, "Das Verhaltnis von Mathematik und Ideenlehre bei 
Plato." 

120 Enriques, The Historic Development of Logic, p. 25. 

121 By Augustus De Morgan. See Hill, "Presidential Address on the Theory of Pro- 
portion." 

122 See Vogt, "Die Lebenzeit Euklids." 

123 Brunschvicg, Les £tapes de la philosophie mathematique, pp. 84-85. 

124 T. L. Heath, History of Greek Mathematics, I, 356. 



46 Conceptions in Antiquity 

pragmatic utilitarianism. For him it had entered the domain of logic, 
and in this connection Proclus tells us that Euclid subjected to rigor- 
ous proofs what had been negligently demonstrated by his prede- 
cessors. 126 Nevertheless, the Elements retained the realism which was 
so clearly apparent in Aristotelian logic. 126 

Although Aristotle had rejected Plato's doctrine of ideas, he had 
retained a belief in a natural order of science and in the necessary 
character of principles. This latter confidence was adopted likewise by 
Euclid. Greek geometry was not formal logic, made up of hypothetical 
propositions, as mathematics largely is today; but it was an idealized 
picture of the world of actuality. Just as Aristotle seems not to have 
clearly recognized the tentative character of scientific knowledge (thus 
leaving himself open to the attacks of the Skeptics), so also he failed 
to appreciate that although the conclusions drawn by mathematics 
are necessary inferences from the premises, nevertheless the latter are 
quite arbitrarily selected, subject only to an inner compatibility. 
Aristotle considered hypotheses and postulates as statements which 
are assumed without proof, but which are nevertheless capable of 
demonstration. 127 Although he admitted that "we must get to know 
the primary premises by induction" (rather than by pure intellection, 
as Plato had believed), he maintained that "since except intuition 
nothing can be truer than scientific knowledge, it will be intuition 
that apprehends the primary premises," and primary premises are 
therefore "more knowable than demonstrations." 128 

The Euclidean view was similar to the Peripatetic attitude in giving 
to geometry the characteristic form of logical conclusions from neces- 
sary postulates. As such, it excluded any notions the nature of which 
was not clearly and compellingly "felt" through intuition. The infinite 
was never invoked in the demonstrations, true to Aristotle's statement 
that it was unnecessary, its place being taken by the method of 
exhaustion which had been developed by Eudoxus. The limitation of 
the concept of number to that of positive integers apparently was 

115 Cf . Proclus Diadochus, In primum Euclides elementorum librum commentariorum . . . 
ibri IIII, p. 43. 

m Enriques, The Historic Development of Logic, p. 25; cf. also Burtt, Metaphysical Foun- 
dations of Modem Physical Science, p. 31. 

m Analytica posterior a I. 76b. 

m Analytica posterior a II. 100b; cf. also Brunschvicg, Les fttapes de la philosophic 
mathlmatique, pp. 86-93. 



Conceptions in Antiquity 47 

continued, a broader view being made unnecessary by the Eudoxian 
theory of proportion. 129 

For Euclid ratio was not a number in the abstract arithmetical 
sense (and in fact it did not become so until the time of Newton), 130 
and the treatment of the irrational in the Elements is completely geo- 
metrical. Furthermore, the axioms, postulates, and definitions of 
Euclid are those suggested by common sense, and his geometry never 
loses contact with spatial intuition. 131 His premises are the dictates 
of sensory experience, much as Aristotle's science may be charac- 
terized as a glorification of common sense. Such purely formal, logical 
concepts as those of the infinitesimal and of instantaneous velocity, of 
infinite aggregates and the mathematical continuum, are not elab- 
orated in either Euclidean geometry or in Aristotelian physics, for 
common sense has no immediate need for them. The ideas which were 
to lead to the calculus had not in Euclid's time reached a stage' at 
which a logical basis could have been afforded; mathematics had not 
attained the degree of abstraction demanded for symbolic logic. 
Although the origin of the notions of the derivative and the integral 
are undoubtedly to be found in our confused thought about varia- 
bility and multiplicity, the rigorous formulation of the concepts in- 
volved, as we shall find, demanded an arithmetical abstraction which 
Euclid was far from possessing. Even Newton and Leibniz, the inven- 
tors of the algorithmic calculus, did not fully recognize the need for 
it. The logical foundations of the calculus are much further removed 
from the vague suggestions of experience — much more subtle — than 
those of Euclidean geometry. Since, therefore, the ideas of variability, 
continuity, and infinity could not be rigorously established, Euclid 
omitted them from his geometry. The Elements are based on "refined 
intuition," 132 and do not allow free scope to the "naive intuition" 
which was to be especially active in the genesis of the calculus in the 
seventeenth century. 133 

From the point of view of the development of the calculus, there- 

m See Stolz, Vorlesungen fiber allgemeine Arithmetik, I, 94; cf. also Schubert, "Principes 
fondamentaux de l'arithmetique," pp. 8-9. 

130 See Newton, Opera omnia, I, 2. 

131 Cf. Barry, The Scientific Habit of Thought, pp. 215-17. 

m As Felix Klein (The Evanston Colloquium Lectures on Mathematics, pp. 41-42) so aptly 
puts it. 
m Ibid. 



48 Conceptions in Antiquity 

fore, the Elements of Euclid show an uninteresting inflexibility of 
rigor, discouraging to the growth of such new speculations and dis- 
coveries. The work of Euclid represents the final synthetic form of all 
mathematical thought — the elaboration by deductive reasoning of the 
logical implications of a set of premises. Back of his geometry, how- 
ever, stood several centuries of analytical investigation, carried out 
often on the basis of empirical research, or on uncritical intuition, or, 
not infrequently, on transcendental speculation. It was to be largely 
from indagation of a similar type, rather than from the rigorously 
precise thought of Euclid, that the development of the concepts of 
the calculus was to proceed. This in its turn was necessarily to give 
way, in the nineteenth century, to a formulation as eminently deduc- 
tive — albeit arithmetic rather than geometric — as that found in the 
Elements. 

The greatest mathematician of antiquity, Archimedes of Syracuse, 
displayed two natures, for he tempered the strong transcendental 
imagination of Plato with the meticulously correct procedure of 
Euclid. He "gave birth to the calculus of the infinite conceived and 
brought to perfection successively by Kepler, Cavalieri, Fermat, 
Leibniz, and Newton," 134 and so made the concepts of the derivative 
and the integral possible. In the demonstration of his results, how- 
ever, he adhered to the clearly visualized details of the Eudoxian 
procedure, modifying the method of exhaustion by considering not 
only the inscribed figure but the circumscribed figure as well. The 
deductive method of exhaustion was not a tool well adapted to the 
discovery of new results, but Archimedes combined it with infini- 
tesimal considerations toward which Democritus and the Platonic 
school had groped. The freedom with which he handled these is 
shown most clearly in the treatise to which we have already referred, 
the Method.™ 

This work, addressed to Eratosthenes the geographer, astronomer, 
and mathematician of Alexandria, was lost and remained largely un- 

134 Chasles, Aperqu historique sur Vorigine et le developpement des methodes en ge'ome'trie, 
p. 22. 

135 For the works of Archimedes in general, see Heiberg, Archimedis opera omnia and 
T. L. Heath, The Works of Archimedes. For Archimedes' Method, see T. L. Heath, The 
Method of Archimedes, Recently Discovered by Heiberg; Heiberg and Zeuthen, "Eine neue 
Schrift des Archimedes"; and Smith, "A Newly Discovered Treatise of Archimedes." 



Conceptions in Antiquity 



49 



known until rediscovered in 1906. In it Archimedes disclosed the 
method which is presumably that which he employed in reaching 
many of his conclusions in problems involving areas and volumes. 
Realizing that it is advantageous to have a preliminary notion of the 
result before carrying through a deductive geometrical demonstration, 
Archimedes employed for this purpose, in conjunction with his law of 
the lever, the idea of a surface as made up of lines. For example, he 
showed that the truth of the proposition that a parabolic segment is 
£ the triangle having the same base and vertex (the vertex of the 
segment being taken as the point from which the perpendicular to the 
base is greatest) is indicated by the following considerations from 




FIGURE 2 



mechanics. 136 In the diagram given in Figure 2, in which V is the 
vertex of the parabola, BC is tangent at B, BD = DP, and X is any 
point on AB, we know from the properties of the parabola that for 

•*• *Y U .U .- XX '" AB BD DP ^ 

any position of X we have the ratio = = = . But 

XX' AX DX" DX" 

X" is the center of gravity of XX'", so that from the law of the lever 

we see that XX', if brought to P as its midpoint, will balance XX'" in 

its present position. This will be true for all positions of X on AB. 

Inasmuch as the triangle ABC consists of the straight lines XX'" in 

this triangle, and since the parabolic segment A VB is likewise made 

"• See T. L. Heath, The Method of Archimedes, Proposition I, pp. 15-18. 



50 Conceptions in Antiquity 

up of the lines XX', we can conclude that the triangle ABC in its 
present position will be in equilibrium at D with the parabolic seg- 
ment when this is transferred to P as its center of gravity. But the 
center of gravity of ABC is on BD and is \ the distance from D to B, 
so that the segment A VB is \ the triangle A BC, or i the triangle A VB. 

This method of Archimedes indicates an anticipation of the use of 
the concept of the indivisible which was to be made in the fourteenth 
century and which, when developed again more freely in the seven- 
teenth century, was to lead directly to the procedures of the calculus. 
The basis of the method is to be found in the assumption of Archimedes 
that surfaces may be regarded as consisting of lines. We do not know 
in precisely what sense he intended this to be understood, for he did 
not speak of the number of elements in each figure as infinite, but 
said rather that the figure is made up of all the elements in it. That he 
probably thought of them as mathematical atoms is indicated not 
only by this manner of expression, but also by the highly suggestive 
fact that he was led to many new results by a process of balancing, in 
thought, elements of dissimilar figures, using the principle of the 
lever precisely as one would in weighing mechanically a collection of 
thin laminae or material strips. 

Using this heuristic method, Archimedes was able to anticipate the 
integral calculus in achieving a number of remarkable results. He 
discovered, among other things, the volumes of segments of conoids 
and cylindrical wedges and the centers of gravity of the semicircle, of 
parabolic segments, and of segments of a sphere and a paraboloid. 137 
However, to assert that here "for the first time one can correctly speak 
of an integration" 138 is to misinterpret the mathematical process 
known by this name. The definite integral is defined in mathematics 
as the limit of an infinite sequence and not as the sum of an infinite 
number of points, lines, or surfaces. 139 Infinitesimal considerations, 
similar to those in the Method, were at a later period to furnish per- 
haps the strongest incentive to the development of the calculus, but, 

m See The Works of Archimedes, Chap. VII, "Anticipations by Archimedes of the Inte- 
gral Calculus"; also Method. 

m Hoppe, "Zur Geschichte der Infinitesimalrechnung," p. 154; cf. also p. 155. 

w T. L. Heath (The Method of Archimedes, pp. 8-9) correctly points out that the method 
here used is not integration; but he gratuitously imputes to Archimedes the concept of the 
differential of area. 



Conceptions in Antiquity 51 

as Archimedes realized, they lacked in his time all basis in rigorous 
thought. This they continued to do until the concepts of variability 
and limit had been carefully analyzed. For this reason Archimedes 
considered that this method merely indicated, but did not prove, 
that the result is correct. 140 

Archimedes employed his heuristic method, therefore, simply as an 
investigation preliminary to the rigorous demonstration by the method 
of exhaustion. It was not a generous gesture that led Archimedes to 
supplement his "mechanical method" by a proof of the results in the 
rigorous manner of the method of exhaustion; it was, rather, a mathe- 
matical necessity. It has been asserted that Archimedes' method 
"would be quite rigorous enough for us today, although it did not 
satisfy Archimedes himself." 141 Such an assertion is strictly correct 
only if we ascribe to him our modern doctrines on number, limit, and 
continuity. This ascription is hardly warranted, inasmuch as Greek 
geometry was concerned with form rather than with variation. It was, 
as a result, necessarily unable to frame a satisfactory definition of the 
infinitesimal, which of necessity was to be regarded as a fixed quantity 
rather than as an auxiliary variable. Archimedes was probably well 
aware of the lack of any sound basis for his method and for this reason 
recast all of his analysis by infinitesimals in the orthodox synthetic 
form, much as Newton was to do almost nineteen hundred years later 
after the methods of the calculus had been discovered but still lacked 
adequate foundation. 

The suggestive analysis of the problem of determining the area of 
a parabolic segment had been given by Archimedes in the Method. 
However, formal proofs (both mechanical and geometrical) of the 
proposition were carried out by the method of exhaustion in another 
treatise, the Quadrature of the Parabola.™ In these proofs Archimedes 
followed his illustrious predecessors in omitting all reference to the 
infinite and the infinitesimal. In the geometrical demonstration, for 
example, he inscribed within the parabolic segment a triangle of area 
A, having the same base and vertex as the segment. Then within each 

140 T. L. Heath, History of Greek Mathematics, II, 29. 

141 T. L. Heath, The Method of Archimedes, p. 10. 

142 See The Works of Archimedes, and T. L. Heath, History of Greek Mathematics, II, 85- 
91. A good adaptation of the geometrical proof is given in Smith, History of Mathematics, 
II, 680-83. 



52 Conceptions in Antiquity 

of the two smaller segments having the sides of the triangle as bases, 
he similarly inscribed triangles. Continuing this process, he obtained a 
series of polygons with an ever-greater number of sides, as illustrated 
(fig. 3). He then demonstrated that the area of the nth such polygon 
was given by the series A (l + 4 + ^2 + . . . + 4»-i), where A 
is the area of the inscribed triangle having the same vertex and base 
as the segment. The sum to infinity of this series is %A, and it was prob- 
ably from this fact that Archimedes inferred that the area of the para- 
bolic segment was also §A , 143 

However, he did not state the argument in this manner. Instead of 
finding the limit of the infinite series, he found the sum of n terms and 
added the remainder, using the equality 

A i 1 + I + ? + ••• + I«-i + \ • i»-i) = \ A - 




FIGURE 3 

As the number of terms becomes greater, the series thus "exhausts" 
$A only in the Greek sense that the remainder, j(^n-ijA, can be 

made as small as desired. This is, of course, exactly the method of 
proof for the existence of a limit, 144 but Archimedes did not so inter- 
pret the argument. He did not express the idea that there is no re- 
mainder in the limit, or that the infinite series is rigorously equal 
to iA. m Instead, he proved, by the double reductio ad absurdum of 
the method of exhaustion, that the area of the parabolic segment 
could be neither greater nor less than iA . In order to be able to define 
$A as the sum of the infinite series, it would have been necessary to 
develop the general concept of real number. Greek mathematicians 
did not possess this, so that for them there was always a gap between 
the real (finite) and the ideal (infinite). 

143 T. L. Heath, "Greek Geometry with Special Reference to Infinitesimals." 

144 As Miller pointed out in "Some Fundamental Discoveries in Mathematics." 
146 The Works of Archimedes, p. cxliii. 



Conceptions in Antiquity 53 

It is not strictly correct, therefore, to speak of Archimedes' geo- 
metrical procedure as a passage to the limit, for the essential part of 
the definition of a limit is the infinite sequence. 146 Inasmuch as he did 
not invoke the limit concept, it is hardly correct to say that in finding 
the sum of such series Archimedes answered in a very explicit and 
definite manner some of the difficult questions raised by Zeno, and 
that "These difficulties were completely solved by the Greek mathe- 
maticians, and further serious arguments along this line seem to be 
based upon ignorance or perversity." 147 The notion of the limit of an 
infinite series is essential for the clarification of the paradoxes; but 
Greek mathematicians (including Archimedes) excluded the infinite 
from their reasoning. The reasons for this ban are obvious: intuition 
could at the time afford no clear picture of it, and it had as yet no 
logical basis. The latter difficulty having been removed in the nine- 
teenth century and the former being now considered irrelevant, the 
concept of infinity has been admitted freely into mathematics. The 
related limit concept is now invoked in the explication of the para- 
doxes, as well as in a simplification of Archimedes' long indirect 
demonstrations. 

The series given above is not the only one found in Archimedes' 
work. In determining, by the method of exhaustion, the volume of a 
segment of a paraboloid of revolution, he was led to investigations of 
a similar nature. A consideration, in some detail, of the application 
of the method which Archimedes here made 148 may be desirable at 
this point, in order to bring out clearly the general character of his 
procedure and to point out to what a remarkable extent it resembles 
that used in the integral calculus, even though Archimedes did not 
explicitly employ the limit concept. 

Archimedes first circumscribed about the solid ABC (which he 

called a conoid) the cylinder ABEF (fig. 4) having the same axis, 

CD, as has the paraboloidal segment. He then divided the axis into 

n equal parts of length h and through the points of division passed 

planes parallel to the base. On the sections of the paraboloid thus 

formed he constructed inscribed and circumscribed cylinder frusta, as 

"• C. R. Wallner, "Ober die Entstehung des Grenzbegriffes," p. 250. See also Hankel, 
"Grenze"; and Wieleitner, Die Geburt der Modemen Mathematik, II, 12. 

147 Miller, "Some Fundamental Discoveries," p. 498. 

148 On Conoids and Spheroids, Propositions 21, 22; The Works of Archimedes, pp. 131-33. 



54 



Conceptions in Antiquity 



shown in the figure. He was then able to establish the equivalent of 
the proportions: 

Cylinder ABEF nVt 

Inscribed figure h -\- 2h + . . . + (n — \)h 

Cylinder ABEF n% 

and — 



Circumscribed figure h + 2h + . . . + nh 
Archimedes had previously shown 149 (using a method cast in geo- 
metrical form but otherwise much like that ordinarily employed in 
elementary algebra to determine the sum of an arithmetic progression) 
that h + 2h + ... + («- \)h < %n 2 h and that h + Ik + . . . + 
nh > \rfih. At this point modern mathematics would employ the limit 

C p 




D 
FIGURE 4 



to become infinite, 
= 2. Not so 



concept and, allowing the series h + 2h -f 3h + 

i • / ffth 

would conclude that um ( 

»-»- \h + 2k + 3h + . . . + nh, 

Archimedes. Instead of doing this, he showed that the proportions 

above may be written, as a result of these inequalities, 

Cylinder ABEF 



Cylinder ABEF ^ 2 , 
> - and 
1 



2 
< -. 

1 



inscribed figure 1 circumscribed figure 

Now by the principle of exhaustion and the usual reductio ad absur- 
dum, he concluded that the paraboloidal segment can be neither 
greater than, nor less than, half the cylinder ABEF. 

In the argument of Archimedes, the logical validity of a conclusion 

149 In Proposition 10 of the work On Spirals. See Works of Archimedes, pp. 163-65. 



Conceptions in Antiquity 55 

based on the numerical concept of the limit of an infinite series was 
not admitted, but was replaced by one based on the rigorous geomet- 
rical method of exhaustion. It has been said that the differences in 
the methods of infinitesimals, of exhaustion, and of limits are felt to 
be more in the words than in the ideas; 150 but such an assertion may 
lead to serious misinterpretation. The methods are, of course, inter- 
related, and in consequence they lead to identical results; but the 
points of view are distinctly different, as we have seen. Although in a 
broad sense the procedures of Archimedes may be considered as 
"practically integrations," 151 or even as representing in a general way 
"an integration process," 152 it is indeed far from correct to speak of 
.any one of them as a "veritable integration" 153 or as "the equivalent 
of genuine integration." 154 The definite integral requires for its correct 
formulation an appreciation of the notions of variability and func- 

n 

tionality, the formation of the characteristic sum 2 f(x { ) Ax i} and the 

»=i 

application of the concept of the limit of the infinite sequence obtained 
from this sum by allowing n to increase indefinitely as Ax { becomes 
indefinitely small. These essential aspects of the integral are, of 
course, at no place indicated in the work of Archimedes, for they were 
extrinsic to the whole of Greek mathematical thought. 

The series h -f- 2h + 3h + . . . , employed by Archimedes in the 
proposition above, was to figure prominently in the gropings toward 
the calculus in the seventeenth century, so that it may be well to 
point out that this geometrical demonstration is in broad outline 
equivalent to performing the integration indicated by j^xdx. In 
determining that the area bounded by the polar axis and by one 

turn of the spiral P = — is £ that of the circle of radius a, 155 Archimedes 

had occasion to make a similar calculation, equivalent to evaluating 
j%x?dx. This was not done directly and arithmetically by determining 

lim A 2 + ( 2h ) 2 + • • • + (nh) 2 \ ^ , 

lim ( L^_^! — ! — L_^ — L to be £, as he might easily 

150 See Milhaud, Nouvelles etudes, p. 149. 

161 T. L. Heath, History of Greek Mathematics, II, 3. 

m Karpinski, "Is There Progress in Mathematical Discovery?" p. 48. 

163 Zeuthen, Geschichte der Mathematik im Altertum und Mittelalter, p. 181. 

154 The Works of Archimedes, p. cliii; cf. also p. cxliii. 

m On Spirals, Proposition 24, The Works of Archimedes, pp. 178-82. 



56 Conceptions in Antiquity 

have done had the Greeks not interdicted the infinite, but indirectly 

and geometrically by coupling the inequalities 

ff + (2hY + . . . + (nhY 1 W- + (2/Q 2 + ...+[(»- pffi 

n(nh) 2 n(nh) 2 

with the proof by the method of exhaustion, in a manner similar to 
that employed in the proposition on the paraboloidal segment. 
Archimedes may have known the equivalent result for the sum of 
cubes as well, and much later the Arabs extended his work to include 
also the fourth powers. In the seventeenth century at least a half dozen 
mathematicians — Cavalieri, Torricelli, Roberval, Fermat, Pascal, and 
Wallis — were to extend (all more or less independently) this work of 
Archimedes still further by the determination of ffcfdx for yet other 
values of n, and in this way to point immediately to the algorithm of 
the calculus. The methods which these men were to use were not, in 
general, the careful geometric procedures found in the propositions of 
Archimedes, but were to be based on ideas of indivisibles and of 
infinite series — notions which the decline of the idea of mathematical 
rigor during the intervening years was to make more acceptable to 
mathematicians, even though they were to be at that time no less 
impeachable than in Archimedes' day. 

We have seen that Greek geometry was concerned largely with 
form rather than with variation, so that the function concept was 
not developed. That motion was nevertheless occasionally invoked in 
mathematics is indicated by Plato's suggestion that a line is gener- 
ated by a moving point, as well as by the fact that certain special 
curves, described by a double motion, had been discussed even before 
Plato's time. The most famous of such curves was perhaps the quad- 
ratrix of Hippias, the Sophist. Archimedes may have been influenced 
by Hippias' idea when he denned his spiral as the locus of a point 
which moved with uniform radial velocity along a line, while the line 
in turn revolved uniformly about one of its end points which is kept 
fixed. 156 

In the further study of this curve Archimedes was led, in attempting 
to determine its tangent, to one of the few considerations correspond- 
ing to the differential calculus to be found in Greek geometry. In 
conformity with the static geometry of design, a tangent to a circle 

166 The Works of Archimedes, p. 165. 



Conceptions in Antiquity 57 

had been denned by Euclid 167 as a line touching the circle at only 
one point, and this definition was extended by Greek geometers to 
apply to other curves as well. There is also ascribed to the ancients 158 
the definition, following the suggestion contained in a proposition of 
Euclid, 189 of a tangent as a line touching a curve and such that in 
the space between the straight line and the curve no other straight 
line can be interposed. These definitions were, of course, of restricted 
applicability and did not, in general, suggest a method of procedure 
for drawing tangents. Although Archimedes did not offer a more satis- 
factory definition, he appears, nevertheless, to have employed, for the 
determination of the tangent to his spiral, a method suggestive of a 
more general point of view. As in quadratures he had used considera- 
tions from the science of statics, so here in the problem of tangents 
Archimedes appears to have had recourse to a representation derived 
from kinematics. It seems likely, although Archimedes did not thus 
express the idea, that he found the tangent to the spiral p = ad 
through a determination of the instantaneous direction of motion of 
the point P, by which it is traced. 160 This he probably did by applying 
to the motions by which the spiral may be generated the parallelogram 
of velocities, the principle of which had been perceived by the Peri- 
patetics. 161 The motion of P may be regarded as compounded of two 
resultant motions: one with a radial velocity of constant magnitude 
V r directed along the line OP (see fig. 5), and the other in a direction 
perpendicular to this and having a magnitude which is given by the 
variable product, V a , of the distance OP and the uniform speed of 
rotation. Inasmuch as the distance OP and the speeds are given, the 
parallelogram (in this case a rectangle) of velocities can be con- 
structed and the direction of the resultant velocity, and therefore also 
the tangent PT, determined. 

The determination by Archimedes of the tangent to the spiral has 
been characterized as "a differentiation," 162 or as "corresponding to 

" Book III, Definition 2, in T. L. Heath ed., II, 1. 

168 See, for example, Comte, The Philosophy of Mathematics, pp. 108-10. 

"» Book III, Proposition 16, in T. L. Heath ed., II, 37. 

180 T. L. Heath, History of Greek Mathematics, II, 556-61. 

U1 See Mechanica XXXIII. 854b-855a in The Works of Aristotle, Ross ed., Vol. VI, 
Opuscula; cf. also Duhem, Les Origines de la statique, II, 245; and Mach, The Science of 
Mechanics, p. 511. 

1M Simon, "Zur Geschichte und Philosophic der Differentialrechnung," p. 116. 



58 



Conceptions in Antiquity 



our use of the differential calculus." 163 Such a designation, however, 
is hardly justified. Apparently, Archimedes made no effort to develop 
the idea here involved into a uniform method of attack upon the 
problem of tangents to other curves. When, in the seventeenth cen- 
tury, the method again appeared in the work of Torricelli, Roberval, 
Descartes, and Barrow, the scope of its applicability was somewhat 
extended; but only with the method of fluxions of Newton was there 
presented an algorithmic procedure for determining from the equa- 
tion of any curve a pair of generating motions from which the tangent 
might be found. There was in Greek geometry no idea of a curve as 




FIGURE 5 

corresponding to a function, nor was there a satisfactory definition 
of a tangent in terms of the limit concept. There was therefore in the 
thought of Archimedes no anticipation of the realization that the 
geometrical notion of tangency is to be based upon the function con- 
cept and upon the numerical idea of a limit, i. e., upon the expression 

lim / /(* + *) "/(*) ) wh i ch f urri ishes the basis for the differ- 

ential calculus. There is in the whole of Greek mathematics no clear 
recognition of the need for the limit concept, both for the determina- 
1M T. L. Heath, History of Greek Mathematics, II, 557. 



Conceptions in Antiquity 59 

tion of curvilinear areas and of tangents to curves, and even for the 
very definition of these ideas which intuition vaguely suggests. 

It is therefore incorrect to impute to Archimedes the ideas ex- 
pressed in the integral and the derivative. These notions were not 
part of Greek geometry. Nevertheless, the problems and methods of 
Archimedes furnished probably the strongest incentive to the later 
development of such ideas, for they were leading in precisely this 
direction. The work of Archimedes so strongly suggests the newer 
methods of analysis that in the seventeenth century Torricelli and 
Wallis hazarded the opinion that the ancient Greek mathematicians 
had deliberately concealed under their synthetic demonstrations the 
analytic devices by which they had been led to their discoveries. 164 
Through the discovery of the Method of Archimedes, the assumption 
of the existence of such methods has been proved correct, but the 
failure of ancient geometers further to elaborate these in their works 
is not to be regarded as indicating an intent to deceive. Although 
the notions of instantaneous velocity and of the infinitely small were 
accepted — all too uncritically — by Torricelli, Wallis, and their con- 
temporaries, these ideas were not considered by Greek thinkers as 
admissible in mathematics. In the seventeenth century, however, the 
infinitesimal and kinematic methods of Archimedes were made the 
basis of the differential and the fluxionary forms of the calculus. 
Although Leibniz and Newton were thus to admit into the calculus 
the notions of the infinitely small and of instantaneous velocity, 
these were to remain open to criticism even after that — in fact, until, 
in the nineteenth century, the basic concept, that of the derivative, 
had been carefully defined. 

After the time of Archimedes, the trend of Greek geometry was 
toward applications, rather than toward new theoretical develop- 
ments. 165 No Greek mathematician approached nearer to the calculus 
than had Archimedes. His successors — Hipparchus, Heron, Ptolemy, 
and others — turned to various mathematical sciences, such as as- 
tronomy, mechanics, and optics. Nevertheless, the infinitesimal con- 
siderations of Archimedes were not forgotten by the Greek geometers 
of later times. Toward the end of the third century of our era, in a 

"< T. L. Heath, History of Greek Mathematics, II, 21, 557; Torricelli, Opere, I (Part 1), 140* 
" 5 T. L. Heath, History of Greek Mathematics, II, 198. 



60 Conceptions in Antiquity 

period which is regarded as one of decline in mathematics, the geom- 
eter Pappus not only displayed a familiarity with these methods, 
but was able also to add a new result — known as the Pappus theorem — 
to the work of Archimedes on centers of gravity. 166 

However, further significant advances in geometrical method were 
to be dependent upon certain broad changes in other branches of 
mathematics. One of these was the development of a more highly 
elaborated abstract symbolic algebra; another was the introduction 
into algebra and geometry of the notion of variation — of variables 
and functionality. The Arithmetic of Diophantus, which represents the 
highest development of Greek algebraical thought, was really theo- 
retical logistic, 167 rather than generalized arithmetic or the study of 
certain functions of variables. In it the Peripatetic basis in logic is 
stronger than the Platonic ontological conception of mathematics. 
Only one unknown was introduced in this work, and the only solu- 
tions which were accepted as having meaning were those which, in ac- 
cordance with Aristotelian tradition, could be expressed as the quotient 
of integers. Irrational and, imaginary numbers were not recognized. 168 

The Arithmetic of Diophantus represents a "synocopated algebra," 
in that abbreviations for certain recurring quantities and operations 
are systematically introduced in it. However, in order that his work 
might be associated with geometrical results and later serve as the 
suitable basis for the calculus, it had to be made more completely 
symbolic, the concept of number had to be generalized, and the ideas 
of variable and function had to be introduced. During the Middle 
Ages the interest of the Hindus and the Arabs in algebraic develop- 
ment, and the attack by the Scholastic philosophers upon the prob- 
lems offered by the continuum, were to supply in some measure the 
background required for these changes. When, therefore, in the 
sixteenth and seventeenth centuries, the classical work of Archimedes 
was developed into the methods constituting the calculus, the advance 
was made along lines suggested by the traditions built up during the 
medieval period. 

184 We cannot be sure, however, that the theorem was an original contribution on the 
part of Pappus or that he possessed a proof of it. See Pappus of Alexandria, La Collection 
tnathematique, trans, by Ver Eecke; cf. also Weaver, "Pappus." 

167 Jakob Klein, "Die griechische Logistik und die Entstehung der Algebra"; see also 
Tannery, Pour la science hellene, p. 405. 

188 T. L. Heath, History of Greek Mathematics, I, 462. 



III. Medieval Contributions 



THE MOOT points as to the origin and antiquity of Hindu 
mathematics are not immediately pertinent in a precis of the 
development of the derivative and the integral; for these concepts 
depend on certain logical subtleties, the significance of which appears 
to have surpassed the appreciation, or at least to have escaped the 
interest, of the early Indian mathematicians. 

The Hindus apparently were attracted by the arithmetical and com- 
putational aspects of mathematics, 1 rather than by the geometrical 
and rational features of the subject which had appealed so strongly 
to the Hellenic mind. Their name for mathematics, ganita, meaning 
literally the "science of calculation," 2 well characterizes this prefer- 
ence. They delighted more in the tricks that could be played with 
numbers than in the thoughts the mind could produce, so that neither 
Euclidean geometry nor Aristotelian logic made a strong impression 
upon them. The Pythagorean problem of the incommensurable, which 
was of intense interest to Greek geometers, was of little import to 
Hindu mathematicians, who treated rational and irrational quantities, 
curvilinear and rectilinear magnitudes indiscriminately. 3 With respect 
to the development of algebra, this attitude occasioned perhaps an 
incidental advance, since by the Hindus the irrational roots of quad- 
ratics were no longer disregarded, as they had been by the Greeks, 
and since to the Hindus we owe also the immensely convenient con- 
cept of the absolute negative. 4 These generalizations of the number 
system and the consequent freedom of arithmetic from geometrical 
representation were to be essential in the development of the con- 
cepts of the calculus, but the Hindus could hardly have appreciated 
the theoretical significance of the change. 

Similarly, another consequence of the lack of nice distinction in 
Hindu thought happens to have been responsible for a change which 
corresponds to the modern view. We have seen that the crisis of 

1 Cf. Karpinski, The History of Arithmetic, p. 46. 

2 Datta and Singh, History of Hindu Mathematics, Part I, p. 7. 

3 Lasswitz, Geschichte der Atomistik, I, 185. 

4 Fine, The Number System of Algebra, p. 105. 



62 Medieval Contributions 

the incommensurable led to the abandonment by the Greeks of the 
attempt to associate numbers with all geometrical magnitudes. The 
Pythagorean problem of the application of areas was in general insolu- 
ble in terms of the conceptions of number and geometrical magnitude 
then current. The area of the circle could not literally be exhausted 
by applying rectilinear configurations to it, inasmuch as curvilinear 
magnitudes were fundamentally different. The area of a circle was 
not to be compared with that of a square, for area was not a numerical 
concept, and equality in general — as distinct from equivalence — 
meant congruence. The number * would have had no meaning in 
Greek mathematics. With the Hindus the view was different. They 
saw no essential unlikeness between rectilinear and curvilinear figures, 
for each could be measured in terms of numbers; arithmetic and 
mensuration, rather than geometry and considerations of congruence, 
were fundamental. The strong Greek distinction between the discrete- 
ness of number and the continuity of geometrical magnitude was not 
recognized, for it was superfluous to men who were not bothered by 
the paradoxes of Zeno or his dialectic. Questions concerning incom- 
mensurability, the infinitesimal, infinity, the process of exhaustion, 
and other inquiries leading toward the conceptions and methods of 
the calculus were neglected. 

Operational difficulties were felt in dealing with the number zero — 
difficulties which led Brahmagupta to regard zero as an infinitesimal 
quantity which ultimately reduces to nought; and which caused 
Bhaskara to say that the product of a number and zero is zero, but 
that the number must be retained as a multiple of zero if any further 
operations impend. 5 Yet these difficulties do not appear to have been 
considered with the intention of resolving the logical questions, im- 
plicit in the use of indeterminate forms, which were later to puzzle 
the early users of the calculus. Questions of limits, although implied 
in their work, were not expressly stated. 6 The emphasis which Hindu 
mathematicians placed on the numerical aspect of the subject, to- 
gether with the use of the Hindu numerals and of the principle of 
positional notation (the latter having been employed also by the 
Babylonians) did, of course, make more easily possible the develop. 

5 Datta and Singh, op. cit., p. 242. 

6 Sengupta, "History of the Infinitesimal Calculus in Ancient and Mediaeval India," 
p. 224. 



Medieval Contributions 63 

ment of algebra, and, subsequently, that of the algorithmic procedures 
of the calculus. However, the logical concepts of the derivative and 
the integral are just as easily denned in terms of Greek numeration 
as of our own, so that Hindu mathematics added no thought essential 
to the development of these ideas. 

The Hindu numerals reached Europe through the medium of the 
Arabic civilization. This was preeminently eclectic, so that in Arabian 
mathematics we find both Greek and Hindu elements. The Arabic 
reckoning is based on that of Hindus, and Arabic trigonometry is Indian 
also in its use of the sine and the arithmetic form, rather than of the 
Hipparchan chord and geometric representation. Arabic geometry, 
however, shows the influence of Euclid and Archimedes, and Arabic 
algebra indicates a return to Greek geometric demonstration and the 
Diophantine avoidance of negative numbers. 7 The general character 
of Arabic algebra, however, is somewhat different from that of Pio- 
phantus' Arithmetic, for it is rhetorical rather than syncopated and 
deals mostly with problems of practical life, rather than with the 
abstract properties of numbers. 8 The whole trend of Arabic mathe- 
matics was, like that of the Hindus, directed away from the specula- 
tions on the incommensurable, continuity, the indivisible, and infin- 
ity — ideas which in Greek geometry were leading toward the calculus. 
Furthermore, additions to the classic Greek works — such as the 
treatise we have by Alhazen (or Ibn al-Haitham) on the measurement 
by infinitesimals of the paraboloid and on the summations of the 
cubes and the fourth powers of the positive integers 9 — were slight, 
for Arabic thought lacked the interest which was necessary to pursue 
further such fecund ideas. To the Arabs, however, we owe the preserva- 
tion and transmission to Europe of much of the Greek work which 
would otherwise have been lost. 

Christian Europe had, since the time of Pappus, added practically 
nothing to traditional mathematical theory, and was, in fact, largely 
unfamiliar with the ancient treatises until, in the twelfth century, 
Latin translations were made from Arabic, Hebrew, and Greek 

7 Fine, Number System; cf. also Paul Tannery, Notions historiques, p. 333, and Karpinski, 
Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi, p. 21. 

8 Gandz, "The Sources of al-Khowarizmi's Algebra," pp. 263-77. 

9 See Ibn al-Haitham (Suter), "Die Abhandlung iiber die Ausmessung des Paraboloids." 
Cf. also Wieleitner, "Das Fortleben der archimedischen Infinitesmalmethoden bis zum 
Beginn der 17. Jahr., insbesondere ueber Schwerpunkt bestimmungen." 



64 Medieval Contributions 

manuscripts. Before this time the work of Euclid was known chiefly- 
through the enunciation, largely without proofs, of selected proposi- 
tions given in the early sixth century by Boethius, in his Geometry. 10 
The work of Archimedes fared no better, for by the sixth century the 
only works of his generally known (and commented upon by Eutocius) 
were those on the sphere and cylinder, on the measurement of a circle, 
and on the equilibrium of planes (that is, on the law of the lever). 11 

When in the twelfth and thirteenth centuries the Greek works 
began to appear in Latin translations, they did not meet with an 
enthusiastic reception on the part of European scholars, interested as 
these men were in theology and metaphysics. The interest of Roger 
Bacon and the appearance of works such as those of Jordanus 
Nemorarius show that in the thirteenth century there was no apathet- 
ical lack of mathematical activity; but the knowledge displayed at 
this time indicates an inadequate familiarity with the classic works of 
Greek geometry 12 — a nescience which had led to the designation fuga 
miserorum 13 — for the fifth proposition of Euclid's Elements. 14, 

The inadequate attention paid to Greek geometry during the later 
medieval period was paralleled by a similar lack of zeal for Greek 
and Arabic algebraic methods. The thirteenth century opened aus- 
piciously with the appearance, in 1202, of the Liber abaci of Leonardo 
of Pisa. This, however, was not followed by a comparable work for 
almost 300 years — that is, until the appearance in 1494 of the Summa 
de ariihnietica of Luca Pacioli. The dearth of advances in the mathe- 
matical tradition during this period has been made the basis for 
very severe strictures on the mathematical work done in this inter- 
val. 15 Such condemnation is justified by thinking in terms of contri- 

10 Ball, A Short Account of the History of Mathematics, p. 107. 

11 See The Works of Archimedes, p. xxxv. 

12 Ginsburg, "Duhem and Jordanus Nemorarius," p. 361. 

13 Later pons asinorum. 

M Smith, "The Place of Roger Bacon in the History of Mathematics," pp. 162-67. 

15 Hankel (Zur Geschichte der Mathematik im Alterthum mid Mittelalter, p. 349) says: 
"Mit Erstaunen nimmt man wahr, dass das Pfund, welches einst Leonardo der latein. 
ischen Welt iibergeben, in diesen drei Jahrhunderten durchaus keine Zinsen getragen 
hatte; wir finden, von Kleinigkeiten abgesehen, keinen Gedanken, keine Methode, welche 
nicht aus dem liber abaci oder der practica geometriae bereits wohl bekannt oder ohne 
Weiteres abzuleiten ware." See also pp. 357-58 for a similar complaint. Cajori (A History 
of Mathematics, p. 125) says that in this period "the only noticeable advance is a simplifi- 
cation of numerical operations and a more extended application of them." Similarly, 
Archibald (Outline of the History of Mathematics, p. 27) characterizes the interval from 



Medieval Contributions 65 

butions to Greek geometry 16 and to Arabic algebra (to which, however, 
Nicole Oresme did add in the fourteenth century the conception of a 
fractional power 17 which was later to add to the facility of application 
of the methods of the calculus). During the interval from 1202 to 
1494 there may, indeed, have appeared no successor to either Archi- 
medes or to Leonardo of Pisa. 18 If, on the other hand, we regard the 
broader aspects of mathematics — the speculations and investigations 
which lead up to the propositions which are in the end deductively 
demonstrated — it will appear that this so-called barren period fur- 
nished points of view of significance in the development of the cal- 
culus. In this respect, as in others, there was perhaps as much origi- 
nality in medieval times as there is now. 19 

Throughout the early period of the Middle Ages, Aristotle had 
been known in Europe largely through his logical works. During the 
thirteenth century, however, his scientific treatises circulated freely 
and, although these were condemned at Paris in 1210, 20 their study 
was again established in the university by 1255, at which time nearly 
all of Aristotle was prescribed for candidates for the master's degree. 21 
In the Physica, Aristotle had considered in some detail the infinite, 
the infinitesimal, continuity, and other topics related to mathe- 
matical analysis. These became, particularly in the next century, the 
center of a lively discussion on the part of scholastic philosophers. 
They were studied in the light of Peripatetic philosophy, rather than 
in terms of mathematical postulational thought, but the resulting 
speculations were of service in sustaining an interest in such con- 
ceptions until, at a later date, they became a part of mathematics. 

Leonardo of Pisa to Regiomontanus as "a period of about 250 barren years." Bjornbo 
("Uber ein bibliographisches Repertorium der handschriftlichen mathematischen Literatur 
des Mittelalters," p. 326) says "Von einer Entwickelung in dieser Epoche ist kaum zu 
reden; bedeutende mathematische Fortschritte wird man hier vergebens suchen." 

14 It has been well said by Sarton {Introduction to the History of Science, I, 19-20) that 
we do not do justice to the medievals because we judge their first steps by the Greek last 
steps. 

17 Cf. Algorhmus proportionum, pp. 9-10. Fine {Number System, p. 113) goes so far as 
to say that this is the only contribution of the period to algebra. 

18 Enestrom, "Zwei mathematische Schulen im christlichen Mittelalter." 

19 See Sarton, op. cit., I, 16. 

^Denifle and Chatelain, Ckartularium Universitatis Parisiensis, Vol. I, p. 70, No. 11. 
Cf. also Vol. I, pp. 78-79, No. 20. 

21 Ibid., Vol. I, pp. 277-79, No. 246. Cf. also Rashdall, The Universities of Europe in the 
Middle Ages, I, 357-58. 



66 Medieval Contributions 

One of the best examples of Scholastic thought on these topics is 
that found in the work of a man who exerted a great influence upon 
medieval thought, 22 Thomas Bradwardine, "doctor profundus" and 
Archbishop of Canterbury, and perhaps the greatest English mathe- 
matician of the fourteenth century. In his Geometria speculativaP and 
in the Tractatus de continuo 2i Bradwardine discussed, among other 
things, the nature of continuous magnitude, his view being dominated 
by the Peripatetic opposition to any atomism. The doctrine of Leucip- 
pus and Democritus, which had denied divisibility to infinity, has at 
all times had partisans and adversaries, and the Scholastic period 
was far from exceptional in this respect. The idea of the indivisible, 
during the earlier Middle Ages, was often more elementary than that 
held by Democritus long before. It seems to have been believed by 
Capella, Isidore of Seville, Bede, and others, that time is composed 
of indivisibles, an hour being made up of 22,560 such instants. 25 It 
seems probable, or at least possible, that these instants were regarded 
as atoms of time. During the later medieval period the idea of indi- 
visibles under various forms and modifications was upheld by Robert 
Grosseteste, Walter Burley, and Henry Goethals, among others. 26 On 
the other hand, Roger Bacon protested, in his Opus majus, that the 
doctrine of indivisibles was inconsistent with that of incommensura- 
bility, 27 an argument developed further by Duns Scotus, William of 
Occam, Albert of Saxony, Gregory of Rimini, and others. 28 Brad- 
wardine considered, in the light of the problem of the continuum, the 
divers points of view represented by proponents of the doctrine of 
indivisibles. Some interpreted the question in terms of physical 
atomism, others of mathematical points; some assumed a finite, others 
an infinite, number of points; some postulated immediate contiguity, 
others a discrete set of indivisibles. 29 Bradwardine himself maintained 

22 Duhem, Les Origines de la statiqae, II, 323. 

23 For brief indications of the contents of this work, see Moritz Cantor, Vorlesungen, II, 
103-6; and Hoppe, "Zur Geschichte der Infinitesimalrechnung," pp. 158-60. 

24 For an analysis of this see Stamm, "Tractatus de continuo von Thomas Bradwardina"; 
see also Cantor, Vorlesungen, II, 107-9. 

25 Paul Tannery, "Sur la division du temps en instants au moyen age," p. 111. 

24 Stamm, "Tractatus de continuo," pp. 16-17; cf. also Duhem, Etudes sur Leonard de 
Vinci, II, 10-18. 

27 Smith, The Place of Roger Bacon in the History of Mathematics, p. 180, n. 

28 Duhem, Etudes sur Leonard de Vinci, II, 8. 

29 Stamm, "Tractatus de continuo," p. 16. 



Medieval Contributions 67 

that continuous magnitudes, although including an infinite number 
of indivisibles, are not made up of such atoms. 30 "Nullum continuum 
ex indivisibilibus infinitis integrari vel componi," said Bradwardine, 31 
using perhaps for the first time in this connection the word which 
Leibniz was to adopt (upon the suggestion of the Bernoulli brothers 32 ) 
to designate in his calculus the sum of an infinite number of infinites- 
imals — the integral. Bradwardine asserted, on the contrary, that a 
continuous magnitude is composed of an infinite number of continua 
of the same kind. The infinitesimal, therefore, evidently possessed for 
him, as for Aristotle, only potential existence. 33 

William of Occam seems to have occupied a position intermediate 
between that of Bradwardine and that held by the supporters of 
indivisible lines. While admitting that no part of any continuum is 
indivisible, he maintained that, contrary to the teachings of Aristotle, 
the straight line does actually (not only potentially) consist of points. 34 
In another connection, however, Occam said that points, lines, and 
surfaces are pure negations, having no reality in the sense that a 
solid is real. 35 One must not, moreover, read too much into the Scho- 
lastic views on the nature of the continuum. The opinion of Brad- 
wardine has been rather freely identified with that of Brouwer and 
the modern intuitionists, who conceive of the continuum as made up 
of an infinite number of infinitely divided continua; 36 the view of 
Occam has been said to correspond to that held by Russell and the 
formalists, who regard the continuum as a perfect set of points every- 
where dense. 37 Such comparisons are justifiable only in a very general 
sense, for the Scholastic speculations invariably centered upon the 
metaphysical question of the reality of indivisibles, rather than upon 
the search for a representation which should be consistent with the 
premises of mathematics. There was in the medieval views no con- 
ception of the rigorous axiomatic foundation of arithmetic which has 

30 Moritz Cantor, Vorlesungen, II, 108. 

3I Moritz Cantor, Vorlesungen, II, 109, n.; Hoppe, "Zur Geschichte der Infinitesimal- 
rechnung," p. 159; cf. also Stamm, op. cit., p. 17. 

32 James Bernoulli, "Analysis problematis antehac propositi," p. 218. 

M Lasswitz, Geschichte der Atomistik, II, 201; Stamm, "Tractatus de continuo," p. 17. 

34 Burns, "William of Ockham on Continuity." 

38 Duhem, £tudes sur Leonard de Vinci, II, 16-17; III, 26. 

38 Stamm, "Tractatus de continuo," p. 20. 

37 See Birch, "The Theory of Continuity of William of Ockham," p. 496. 



68 Medieval Contributions 

characterized modern thought upon this subject. Nevertheless, the 
fourteenth century disputations on the indivisible represent a keen 
appreciation of the difficulties involved and a clarity of thought which 
was, several centuries later, to lend an air of respectability to the 
infinitesimal methods leading to the calculus. 

In discussions of indivisibles, the question of infinite division and 
the nature of the infinite arose, of necessity. In fact, the medieval 
philosophers discussed the question more from the point of view of 
infinite divisibility and infinite aggregates than from that of infinitely 
great magnitudes. Aristotle, it will be recalled, had distinguished two 
kinds of infinity — the potential and the actual. The existence of the 
latter he had categorically denied, and the former he had admitted 
as realized only in cases of infinitely small continuous magnitudes and 
of infinitely large numbers. 38 The Roman poet Lucretius had, with 
keen imagination, upheld the notion of the infinite as indicating more 
than the potentiality of indefinite increase. In focusing attention 
upon infinite multitudes rather than magnitude, he adumbrated a 
number of properties of infinite aggregates, such as that a part may 
in this case be equal to the whole. 39 The work of Lucretius was not, 
however, familiar to the European scholars of the Middle Ages. The 
Aristotelian distinction, on the other hand, was continued by the 
Scholastic philosophers, although with modifications resulting, per- 
haps, from the fact that Christianity recognized an infinite God. In 
the thirteenth century Petrus Hispanus, who became Pope John XXI, 
recognized in his Summulae logicales** two kinds of infinity: a cate- 
gorematic infinity, in which all terms are actually realized, and a 
syncategorematic infinity, which is bound up always with potential- 
ity. 41 This distinction is not greatly different from that which has 
been suggested recently by a mathematician and scientist who would 
discriminate between saying that an infinite aggregate is conceivable 
and saying it is actually conceived. 42 

The discussion of the two infinities, which was begun in the thir- 
teenth century, was continued throughout the fourteenth also. Albert 

nphysica, Book III. 

" See Keyser, "The R6le of the Concept of Infinity in the Work of Lucretius." 

40 Duhem, Etudes sur Uonard de Vinci, H, 22. 

41 Duhem, op. cit., Vol. II, IAonard de Vinci et les deux infinis, pp. 1-53, gives an exten- 
sive account of this question. 

** Enriques, Problems of Science, pp. 127-28. 



Medieval Contributions 69 

of Saxony, for example, brought out the distinction nicely by a mere 
transposition of words, saying that the two views were illustrated 
respectively by the sentences: "in infinitum continuum est divisible," 
and "Continuum est divisible in infinitum." 43 Bradwardine brought 
out the difference, perhaps with less subtlety but certainly more 
clearly, in saying that the categorematic infinity is a quantity with- 
out end, whereas the syncategorematic infinity is a quantity which is 
not so great but that it can be made greater. 44 

Although the distinction between the two infinities was generally 
recognized by the philosophers of the Scholastic period, there were 
significant differences, then as now, on the question of their existence. 
William of Occam, in conformity with his nominalistic attitude and 
with the principle of economy enunciated in his well-known "razor," 
agreed with Aristotle in denying that the categorematic infinity is 
ever realized. Gregory of Rimini, on the other hand, maintained 45 
what mathematics in the nineteenth century was to demonstrate: 
that there is in thought no self-contradiction involved in the idea of 
an actual infinity — the so-called completed infinite. 

More interesting from the mathematical point of view than these 
philosophical and discursive discussions are the remarks made on the 
subject of the infinite by Richard Suiseth, popularly known as the 
Calculator, in his Liber calculationum. This was composed later than 
1328, inasmuch as it refers to Bradwardine's 46 Liber de proportionibus 
of that year, 47 and probably dates from the second quarter of the 
fourteenth century. 48 Although more interested in dialectical argu- 

43 Duhem, Etudes sur Leonard de Vinci, II, 23. 

44 Stamm, "Tractatus de continuo," pp. 19-20. 

46 See Duhem, Etudes sur Leonard de Vinci, II, 399-401. 

46 Duhem {Etudes sur Leonard de Vinci, III, 429) has incorrectly asserted that Calculator 
names in his work only Bradwardine, Aristotle, and Averroes, but there are in the Liber 
calculationum references also to ancient mathematicians, for example, Euclid (fol. 29 r , 
cols. 1-2) and Boethius (fol. 43", col. 1). 

47 "Ut venerabilis magister Thomas de berduerdino in suo libro de proportionibus 
liquide declarat." Liber calculationum, fol. 3", col. 1. Professor Lynn Thorndike has kindly 
allowed me to make use of his rotograph of a copy in the British Museum, I B. 29, 968, 
fol. l r — 83.* This is a copy of the undated editio princeps at Padua, placed by Thorndike 
(A History of Magic and Experimental Science, III, 372) in the year 1477. Inasmuch as 
this work is neither well known nor readily available, passages from it will be cited at 
some length. The printed text contains so many abbreviations that it is difficult to read, 
but transcriptions from it will be given in full. 

48 See Thorndike, History of Magic, III, 375. Stamm ("Tractatus de continuo," p. 24) 
would place its appearance after 1350. 



70 Medieval Contributions 

ments and subtle sophisms concerning infinity than in its adequate 
definition, Suiseth made several comments on the subject which are 
of particular mathematical significance. In the second chapter he said 
that all sophisms regarding the infinite could be easily resolved by 
recognizing that a finite part can have no ratio to an infinite whole. 49 
This conclusion, he said, would be conceded by the imagination, for 
the contrary would imply that any part, when added to the whole, 
would not change it in magnitude. Arguments with respect to the 
infinite do not proceed, therefore, as do those concerning finite 
quantities. 50 

About two hundred years later Galileo remarked still more clearly 
the essential difference between the rules for the finite and those for 
the infinite, but he focused attention upon the correspondence between 
infinite aggregates, rather than upon the ratio of finite to infinite 
magnitudes — a change of view which led to the final formulation of the 
calculus in the nineteenth century. There is in the statements of 
Calculator a cogency and a warning which might well have been 
observed in later centuries, but they display the Peripatetic propensity 
to regard the infinite as a magnitude, rather than as an aggregation 
of terms. The fruitfulness of the infinite in the work of Archimedes 
arose out of the infinite collections of lines or of terms in a series. 
Reference will be made below, to be sure, to the study by Calculator 
of an infinite series, but in this connection it will be seen that it was 
not the endlessness of the sequence of terms which most interested 
him, but a certain infinite magnitude. A consideration of this series 
will require, however, the study of a larger problem with which it was 
associated, and to which we shall now turn. 

The blending of theological, philosophical, mathematical, and sci- 
entific considerations which has so far been evident in Scholastic 
thought is seen to even better advantage in a study of what was per- 

49 "Infinite quasi sophismata possunt fieri de infinite que omnia si diligenter inspexeris 
quod nullius partis ad totum infinitum est aliqua proportio faciliter dissoluere poteris per 
predicta." Liber calculationum, fol. 8", col. 2. 

60 "Que potest concedi de inmaginatione et causa est quia nulla pars finite finite intensa 
respectu tocius infiniti aliquid confert quia nullam habet proportionem ad illud infini- 
tum si tamen subiectum esset finitum conclusio non foret inmaginabilis quia tunc conclusio 
inmediate repugnaret illi positioni quia tunc quelibet pars in comparatione ad totum 
conferet aliquantum et sic non est nunc ideo nullum argumentum proceditur de infinite 
sicut faceret de finite." Ibid., fol. 8 r , col. 2, fol. 8 s , col. 1. 



Medieval Contributions 71 

haps the most significant contribution of the fourteenth century to 
the development of mathematical physics. It has commonly been 
protested that additions, if any, which were made to scientific knowl- 
edge in the medieval period lay solely in the field of practical dis- 
coveries and applications; and that the only mathematical achieve- 
ment during this time was the simplification of the rules of operation 
for the Hindu-Arabic numerals, the latter having been made known 
in Europe by Leonardo of Pisa and other men of the thirteenth cen- 
tury. There is at least one exception to such an assertion, for it was 
precisely during this interval, and particularly in the fourteenth cen- 
tury, that a theoretical advance was made which was destined to be 
remarkably fruitful in both science and mathematics, and to lead in 
the end to the concept of the derivative. This consisted in the idea — 
often expressed, to be sure, in terms of dialectical rather than mathe- 
matical method — of studying change quantitatively, and thus ad- 
mitting into mathematics the concept of variation. 51 

Heraclitus, Democritus, and Aristotle had made some qualitative 
metaphysical speculations on the subject of motion, and occasionally 
Greek geometers (Hippias, Archimedes, Nicomedes, Diodes) had al- 
lowed this notion to enter their thoughts (though not their proofs); 
but the idea of representing continuous variation by means of geo- 
metrical magnitude or of studying it in terms of the discreteness of 
number does not seem to have arisen with them. The Greek sciences 
of astronomy, optics, and statics had all been elaborated geometrically, 
but there was no such representation of the phenomena of change. 
Archimedes' famous work in statics was not paralleled by any equiv- 
alent kinematic system which admitted of representation in the form 
of mathematical propositions. 

Perhaps the demand for rigor in Greek thought, which made con- 
gruence fundamental in geometry and which allowed no confusion of 

51 It is preposterous to assert — as does Tobias Dantzig (Aspects of Science, p. 45) — 
that, with reference to Aristotle and the Schoolmen, "his casuistry, his predilection for the 
static, his aversion for everything that moved, changed, flowed, or evolved admirably 
suited their purposes." A brief examination of the works of Aristotle on physical science 
and a glance at A Catalogue of Incipits of Mediaeval Scientific Writings in Latin by Thorn- 
dike and Kibre will show how untrue is such a statement. Aristotle made extensive investi- 
gations, from the point of view of physics, into the phenomena of motion. Scholastic 
philosophers not only continued his work, but also added the quantitative form of ex- 
pression which was so successfully developed later in the seventeenth century. 



72 Medieval Contributions 

the discrete with the continuous, could not be met by any attempt to 
establish a science of dynamics. Moreover, Greek astronomy lacked 
the concept of acceleration; the motions involved were all uniform 
(and hence eternal) and circular, and could, in this case, be represented 
by the geometry of the circle. No such uniformity was apparent for 
local motion — that is, for terrestrial changes of position. Motion was, 
it appeared, a quality rather than a quantity; and there was among 
the ancients no systematic quantitative study of such qualities. 82 
Aristotle had spoken of mathematics as concerned with "things 
which do not involve motion," and had held that mathematics 
studies objects qua continuous, physics qua moving, and philosophy qua 
being. 63 In general Greek mathematics was the study of form, rather 
than variability. The quantities entering into Diophantine algebraic 
equations are constants, rather than variables, and this is true also 
of Hindu and Arabic algebra. In the Scholastic period, however, there 
arose a problem which was ultimately to change this view. Aristotle 
had considered motion a quality that does not increase and decrease 
through the joining together of parts, as does a quantity, 64 and this 
idea dominated most thought until toward the end of the thirteenth 
century, 66 at which time a reaction against Peripateticism arose, which 
was to lead, at Paris, to new views on the subject of motion. 66 

Bacon, in the second half of the thirteenth century, still followed 
the Aristotelian discussion of motion, 67 but a new approach to the prob- 
lem was evidenced early in the fourteenth century by the introduction 
of the idea of impetus — the notion that a body, once set in motion, 
will continue to move because of an internal tendency which it then 
possesses, rather than, as Peripatetic doctrine had taught, because of 
the application of some external force, such as that of the air, which 
continues to impel it. This doctrine, which has been ascribed to Jean 
Buridan, 68 was particularly significant as an adumbration of the 
famous work in dynamics of Galileo almost three centuries later. At 

52 Brunschvicg, Les Stapes de la philosophic mathe'matique, p. 97. 
w Physica II. 198a; Metaphysica 1061. 

84 Duhem, Eludes sur Ltonard de Vinci, III, 314-16; cf. also Wieleitner, "Ueber den 
Funktionsbegriff und die grapbische Darstellung bei Oresme," pp. 196-97. 
56 Wieleitner, "Ueber den Funktionsbegriff," p. 197. 
'• Duhem, Etudes sur Leonard de Vinci, II, pp. iii-iv. 

67 Thomson, "An Unnoticed Treatise of Roger Bacon on Time and Motion." 
58 See Duhem, Etudes sur Ltonard de Vinci, passim. 



Medieval Contributions 73 

the time of its inception it served also to make more acceptable the 
intuitive notion of instantaneous velocity, an idea excluded by 
Aristotle from his science, but implied by the quantitative study of 
variation of the fourteenth century. At that stage no precise definition 
of instantaneous rate of change could, of course, be given — nor was 
one given by Galileo — but there appeared at the time a large number 
of works, more philosophical than mathematical, all based on the 
intuition of this concept which everyone thinks he possesses. These 
tractates were devoted to a discussion of the latitude of forms, that 
is, of the variability of qualities. 

There seems to be no scientific term which correctly expresses the 
equivalent of the word form as here used. It refers in general to any 
quality which admits of variation and which involves the intuitive 
idea of intensity — that is, to such notions as velocity, acceleration, 
density. These concepts are now expressed quantitatively in terms of 
limits of ratios — that is, simply as numbers — so that no need is now 
felt for a word to express the medieval idea of a form. In general, the 
latitude of a form was the degree to which the latter possessed a cer- 
tain quality, and the discussion centered about the intensio and the 
remissio of the form, or the alterations by which this quality is ac- 
quired or lost. Aristotle had distinguished between uniform and non- 
uniform velocity, but the critical dialectical treatment of the Scho- 
lastics went much further. In the first place, the time rates of change 
which they considered were not necessarily those of distance, but 
included many others as well, such as those of intensity of illumina- 
tion, of thermal content, of density. Secondly — and more signifi- 
cantly — they distinguished not only between latitudo uniformis and 
latitudo difformis (that is, between uniform and nonuniform rates of 
change), but proceeded further to classify the latter as either latitudo 
uniformiter difformis or latitudo difformiter difformis (that is, according 
as the instantaneous rate of change of the rate of change was uniform 
or not); and the last-mentioned were sometimes in turn further 
divided into either latitudo uniformiter difformiter difformis or latitudo 
difformiter difformiter difformis. 

These attempts to introduce order, by means of verbal arguments, 
into the disconcerting problem of variability were destined to be 
replaced centuries later by equivalent statements, expressed in mathe- 



74 Medieval Contributions 

matics and science by the remarkably concise terminology and sym- 
bolism of algebra and the differential calculus. At the time at which 
they were made, however, they represented the first serious and 
careful effort to make quantitative the idea of variability. 

The origin of the question of the latitude of forms is shrouded in 
doubt. Duns Scotus appears to have been among the first to consider 
the increase and the decrease (intensio and remissio) of forms, 69 
although the loose idea of latitude of forms apparently goes back to 
some time before this, inasmuch as Henry Goethals used the word 
latitudo in this connection. 60 In the early part of the fourteenth cen- 
tury there appeared works on variability and the latitude of forms by 
James of Forli, Walter Burley, and Albert of Saxony. 61 Similar ideas 
appeared also in 1328, at Oxford, in the treatise on proportions of 
Bradwardine. 62 This work is devoted more particularly to mechanics 
than to arithmetic, but the archbishop did not make a special investi- 
gation into the theory of the latitude of forms. 63 

"In the fourteenth century the study of the mathematical sciences 
flourished greatly in Oxford," 64 and it was here that appeared not 
only the work of Bradwardine, but also the "leading model" 66 of such 
treatises on the latitude of forms — the Liber calculationum of Suiseth 
to which reference has already been made. 66 

That the doctrine of the latitude of forms was well known by the 
middle of the fourteenth century is indicated by the fact that Cal- 
culator begins in medias res by a general consideration, in the first 
chapter, of the intension and remission of forms and the question as 
to whether a latitudo difformis corresponds to its maximum or minimum 
gradus or intensity. The sense in which a form could correspond to 
any gradus is not clear, but Calculator appears here to be striving 

59 Von Prantl, Geschichte der Logik im Abendlande, III, 222-23. Cf. Stamm, "Tractatus 
de continuo," p. 23. 

60 Duhem, Etudes sur Leonard de Vinci, III, 314-42. 

61 Wieleitner, "Der 'Tractatus de latitudinibus formarum des Oresme,' " pp. 123-26. 
82 Duhem, Etudes sur Leonard de Vinci, III, 290-301. 

68 Stamm, "Tractatus de continuo," p. 24. 

84 Gunther, Early Science in Oxford, II, 10. 

86 Thorndike, History of Magic, III, 370. 

88 For a general description of the contents of this work, see Thorndike, History of 
Magic, Vol. Ill, Chap. XXIII; and Duhem, Etudes sur Lionard de Vinci, III, 477-81. 
No analysis of the Liber calculationum from the mathematical point of view appears to be 
available. 



Medieval Contributions 75 

toward the idea of average intensity, an idea which could not be 
made precise without the use of the concepts of the calculus. In 
Chapter II, however, he arrived, in this connection, at a result which 
was to be of particular significance in the later development of science 
and mathematics. Here, in considering the intension in difform things, 
he reached the conclusion, in connection with problems on thermal 
content, that the average intensity of a form whose rate of change 
over an interval is constant, or of a form which is such that it is uni- 
form throughout each half of the interval, is the mean of its first and 
last intensities. 67 The rigorous proof of this requires the use of the 
limit concept, but Calculator had resort to dialectical reasoning, based 
on physical experience of rate of change. He argued, in connection 
with such a form, that if the greater intensity is allowed to decrease 
uniformly to the mean while the lesser is increased at the same rate 
to this mean, then the whole is neither increased nor decreased. 68 The 
argument is pursued at further length here, and the author adds, as 
well, numerical illustrations such as the following: If the intensity 
increases uniformly from four to eight, or if for the first half of the 
time it is four, and for the last half it is eight, then the effect is that 
which would result from a uniform intensity of six, operating through- 
out the whole time. 69 

The methods presented here are amplified and applied, in other 
chapters of the Liber calculationum, to questions dealing with density, 
velocity, and the intensity of illumination. In these Suiseth again 
made use of the law that if the rate of change of one of these is con- 
stant, or if the form is such that in each of the two halves of the 

67 "Primo arguitur latitudinem caliditatis uniformiter difformem seu etiam difformem 
cuius utraque medietas est uniformis, et calidum uniformiter difforme suo gradui medio 
correspondere." Liber calculationum, fol. 4", col. 2. 

68 "Capiatur talis caliditas seu tale calidum et remittatur una medietas ad medium et 
intendatur alia ad medium equeuelociter: et sequitur totum non intendi nee remitti; eo 
quod totam latitudinem acquiret. Secundum unam medietatem seu partem sicut deperdet 
secundum aliam partem equalem et in fine erit uniforme sub tali gradu medio. Igitur nunc 
correspondet tali gradui." Ibid. 

69 "Sit enim tale uniformiter difforme seu difforme cuius utraque medietas est uniformis 
una ut. VIII. et alia ut JUL gratia argumenti. Tunc prima qualitas ut .VIII. extenditur 
per medietatem totius per predicta. Ergo solum denominat totum ut quatuor per idem 
prima qualitas ut quatuor per aliam medietatem extensa solum facit ut duo ad totius 
denominatorem. Igitur ille due qualitates totum precise denominabunt ut .VI. qui est 
gradus medius inter illas medietates. Sequitur igitur positio sic in speciali." Ibid., fol. 5*, 
col. 2. 



76 Medieval Contributions 

interval the rate of change is zero, then the average intensity is the 
mean of the first and last values. As was the case in the study of 
variation of the intensity of heat, no definition is given of the terms 
employed, inasmuch as this would presuppose an appreciation of the 
limit concept. The lack of such precise definitions led Calculator into 
difficulties involving the infinite. He considered, for example, a rarity 
of degree zero as a density of infinite degree, and conversely, 70 and 
consequently became involved unnecessarily often in the paradoxes of 
the infinite. 

The tendency displayed by Calculator to consider the infinite from 
the point of view of intensity or magnitude, rather than aggregation, 
is brought out again in the discussion of an example of nonuniform 
variation, which might have had a significant influence upon the 
development of the calculus, had Suiseth's purpose been less that of 
introducing mathematics into dialectical discussions of change and 
more that of bringing the problem of variation into the realm of 
mathematics. Calculator, in the second book of his Liber calculationum, 
had occasion to consider the following problem: if throughout half 
of a given time interval a variation continues at a certain intensity, 
throughout the next quarter of the interval at double this intensity, 
throughout the following eighth at triple this, and so ad infinitum; 
then the average intensity for the whole interval will be the intensity 
of the variation during the second subinterval (or double the initial 
intensity). 71 

This is equivalent to a summation of the infinite series % + f + 

ft 

I + tV + ...+ — +... = 2. It will be recalled that Archimedes 

70 "Ergo .a. est infinite densum et per consequens non est rarum. Ex istis sequitur ista 
conclusio quod aliquid est rarum quod non est rarum quia .a. est rarum quia est uniformiter 
difforme rarum et non est rarum quia est infinite densum. Pro istis negatur utraque con- 
clusio et tunc ad casum positum quod .a. sit unum uniformiter difformiter rarum terminatur 
ad non gradum raritatis negatur casus nam ex quo raritas se habet privative sequitur ut 
argutum est quod ab omni gradu raritatis usque ad non gradum raritatis est latitudo 
infinita quia non gradus raritatis est infinitus gradus densitatis sed impossibile est quod 
aliquid sit uniformiter difforme aliquale mediante latitudine infinita. Ideo casus est im- 
possibilis sicut est impossibile quod aliquid sit uniformiter difforme remissum ad non 
gradum remissionis terminatum." Ibid., fol. 19", cols. 1-2. 

71 "Contra quam positionem et eius fundamentum arguitur sic quia sequitur quod si 
prima pars proportionalis alicuius esset aliqualiter intensa, et secunda in duplo intensior, 
et tertia in triplo intensior, et sic in infinitum totum esset equale intensum precise sicut est 
secunda pars proportionalis quod tamen non uidetur visum." Ibid., fol. 5 , col. 2. 



Medieval Contributions 77 

had made use of certain simple series in connection with his geometry* 
but that the characteristic Greek interdiction of the infinite had led 
him to consider the sum to n terms only. It was apparent to him that 
the series 1 + I + tz + ti + • ■ - + -p-i approached $ in such a 
way that the difference could be made, by taking a sufficiently large 
number of terms, less than any specified quantity. He did not, how- 
ever, go so far as to define i as the "sum" of the infinite series, for this 
would have exposed his thought to the paradoxes of Zeno, unless he 
had invoked the precisely formulated concept of a limit as given in 
the nineteenth century. The Scholastic discussions of the fourteenth 
century, on the other hand, referred frequently to the infinite, both as 
actuality and as potentiality, with the result that Suiseth, with per- 
fect confidence, invoked an infinite subdivision of the time interval 
to obtain the equivalent of an infinite series. He did not resolve the 
aporias of Zeno, to show in what sense an infinite series may be said 
to have a sum — a problem which future mathematicians were to con- 
sider at length. Calculator, instead, was more particularly interested 
in infinite magnitudes than in infinite series. Not only is the time 
interval in his problem infinitely divided, but the intensity itself 
becomes infinite. Now how can a quantity, whose rate of change 
becomes infinite, have a finite average rate of change? Suiseth admitted 
that this paradoxical result was in need of demonstration and so 
furnished at great length the equivalent of a proof of the convergence 
of the infinite series. This he did as follows. 

Consider two uniform and equal rates of change, a and b, operating 
throughout a given time interval, which has been subdivided in the 
ratios \, \, \, . . . . Now let the rate of change b be doubled through- 
out the interval; but in the case of a, let it be doubled in the second 
subinterval; tripled in the third; and so to infinity, as given in the 
problem above. Now the increase in a in the second subinterval, if 
continued constantly throughout this and all following subintervals, 
would result in an increase in the effect equal to that brought about 
by the change in b during the first half of the time. The tripling of a 
in the third subinterval, if continued constantly throughout this and 
the ensuing subintervals, would in turn result in a further increase in 
the effect of a equal to that brought about by the change in b in the 
second subinterval, and so to infinity. Hence the increase resulting 



78 Medieval Contributions 

from the doubling, tripling, and so forth of a is equal to that caused 
by the doubling of b; i. e., the average rate of change in the problem 
considered above is the rate of change during the second subinterval, 
which was to be proved. 72 

The tediously verbose proof given by Calculator is, of course, based 
entirely on arguments appealing to our intuition of uniform rate of 
change. Because Suiseth gave no unambiguously clear definitions of 
velocity, density, intensity of illumination, and other terms used 
freely in his dissertation, his work not infrequently presents — as had 
also, to a certain extent, the Physica of Aristotle 73 — the appearance of 
an effort to propound sophistical questions on the subject of change, 
rather than that of a serious effort to establish a scientific basis for 
the study of the phenomena of motion and variability. 74 

n "Nam apparet quod ilia qualitas est infinita ergo si sit sine contrario infinite denomina- 
bit suum subiectum. Et quod conclusio sequatur arguitur sic: sint .a. .b. duo uniforma 
eodem gradu, et dividatur .b. in partes proportionales et est ilia hora ita quod partes 
maiores terminentur seu incipiant ab hoc instanti, et ponatur quod in prima parte pro- 
portionali illius hore intendatur prima pars .b. ad duplum, et in secunda parte proportionali 
intendatur secunda pars proportionalis illius ad duplum, et sic in infinitum, ita quod in fine 
erit .b. uniforme sub gradu duplo ad gradum nunc habitum. Et ponatur quod .a. in prima 
parte proportionali illius hore intendatur totum residuum .a. prima parte proportionali 
.a. acquirendo totam latitudinem sicut tunc acquireret prima pars proportionalis .b. et in 
secunda parte proportionali euisdem hore intendatur totum residuum .a. a prima parte 
proportionali et secunda illius .a. acquirendo tantam latitudinem sicut tunc acquiret pars 
proportionalis secunda .b. et in tertia parte proportionali indendatur residuum a prima 
parte proportionali et secunda et tertia acquirendo tantam latitudinem sicut tunc acquiret 
tertia pars proportionalis .b. et sic in infinitum scilicet quod quandocumque aliqua pars 
proportionalis .b. intendetur pro tunc intendatur .a. secundum partes proportionales 
subsequentes partem correspondentem in .a. acquirendo tantam latitudinem sicut acquiret 
pars prima in .b. et sint .a.b. consimilia quantitatiue continue quo posito sequitur quod 
.a. et .b. continue equeuelociter intendentur quia .a. continue per partes proportionales 
similiter intendetur sicut .b. quia residuum a prima parte proportionali .a. est equale 
prime parti proportionali eidem. Cum igitur .b. in prima parte proportionali illius hore 
continue intendetur per primam partem proportionalem et .a. per totum residuum a prima 
sua parte proportionali patet quod .a. in prima parte proportionali equeuelociter intendetur 
cum .b. et sic de omni alia parte eo quod quandocumque .b. indendetur per aliquam 
partem proportionalem .a. intendetur per totum interceptum inter partes correspondentes 
sibi et extremum ubi partes terminantur scilicet minores. Cum ergo quelibet pars propor- 
tionalis cuiuslibet continui sit equalis toti intercepto inter eandem et extremum ubi partes 
minores terminantur. Igitur patet quod .a. continue equeuelociter intendetur cum .b. et 
nunc est eque intensum cum .b. ut ponitur in casu. Ergo in fine .a. erit .a. eque intensum 
cum .b. et .a. tunc est tale cuius prima pars proportionalis erit aliqualiter intensa, et 
secunda pars proportionalis in duplo intensior, et tertia in triplo intensior et sic in infini- 
tum. Et .b. erit uniforme gradu sub quo erit secunda pars proportionalis .a. ergo sequitur 
conclusio." Ibid., fol. 5", col. 2— fol. 6 f , col. 1. 

n Cf. Mach, The Science of Mechanics, p. 511. 

m "Multa alia possent fieri sophismata per rarefactionem subiecti, et per fluxum quali- 
tatis et alterations qualitatis si subiectum debet intendi et remitti per huiusmodi rare- 



Medieval Contributions 79 

Suiseth's unfortunate adoption of the Peripatetic attitude that such 
qualities as dryness, coldness, and rarity are the opposites of moist- 
ness, warmth, density — rather than simply degrees of the latter 75 — 
complicated his consideration of these through the unnecessarily fre- 
quent introduction of the paradoxes of the infinite. Nevertheless, to 
Calculator we owe perhaps the first serious effort to make quanti- 
tatively understandable these concepts of mathematical physics. His 
bold study of the change of such quantities anticipated not only the 
scientific elaboration of these, but also adumbrated the introduction 
into mathematics of the notions of variable quantity and derivative. 
In fact, the very words jluxus and jluens, which Calculator used in 
this connection, 76 were to be employed by Newton some three hun- 
dred years later, when in his calculus he spoke of such a variable 
mathematical quantity as a fluent and called its rate of change a 
fluxion. Newton apparently felt as little need as Suiseth for a defi- 
nition of this notion of fluxion, and was satisfied to make a tacit appeal 
to our intuitions of motion. Our definitions of uniform and nonuniform 
rate of change, are, as Suiseth anticipated, numerically expressed; but 
their rigorous definition could be given only after the development, 
to which Newton contributed, of the limit concept. This latter arose 
out of the notions of the calculus, which, in their turn, had evolved 
from the intuitions of geometry. The prolix dialectic of Calculator made 
no appeal to the geometrical intuition, which was to act as an inter- 
mediary between his early attempts to study the problem of varia- 
tion and the final formulation given by the calculus. This link between 
the interminable discursiveness of Suiseth and the concise symbolism 
of algebra was supplied by others of the fourteenth century who 
studied the latitude of forms. Of these the most famous was 
Oresme. 

Nicole Oresme was born about 1323 and died in 1382. He was thus 
somewhat younger than Calculator, from whom we have a manuscript 

factionem, et fluxum et alterationem, ad que omnia considerando proportionem totius ad 
partem responsionem alicere poteris ex predictis faciliter." Liber calculationum, fol. 9 r , col. 
2. The word fluxum has been italicized by me for emphasis. 

76 It should be remarked, in this connection, that several centuries later Galileo entered 
into an extensive discussion as to whether the Peripatetic qualities are to be regarded as 
positive (Le Opere di Galileo Galilei, I, 160 ff.), and that even in the eighteenth and nine- 
teenth centuries the question as to the existence of "frigerific" rays was still argued. See, 
for example, Neave, "J ose Ph Black's Lectures on the Elements of Chemistry," p. 374. 

76 Liber calculationum, fol. 9 r , col. 2; fol. 75", col. 1. 



80 Medieval Contributions 

dated 1337, 77 and his doctrines were in all probability derived from 
those of Suiseth and others of the Oxford School. 78 Furthermore, from 
the very character of their work it would seem reasonable to place the 
composition of the Liber calculationum at Oxford before the appear- 
ance of the work of Oresme at Paris, on the grounds that the latter 
lacks the chief defects of the former — the prolixity of the complicated 
dialectical demonstrations and the author's propensity for subtle 
sophisms. There is in the entire Liber calculationum no diagram or 
reference to geometrical intuition, the reasoning being purely verbal 
and arithmetical. 79 On the other hand, Oresme felt that the multi- 
plicity of types of variation involved in the latitude of forms is dis- 
cerned with difficulty, unless reference is made to geometrical figures. 80 
The work of Oresme therefore makes most effective use of geometrical 
diagrams and intuition, and of a coordinate system, to give his 
demonstrations a convincing simplicity. This graphical representation 
given by Oresme to the latitude of forms marked a step toward the 
development of the calculus, for although the logical bases of modern an- 
alysis have recently been divorced as far as possible from the intuitions 
of geometry, it was the study of geometrical problems and the attempt 
to express these in terms of number which suggested the derivative 
and the integral and made the elaboration of these concepts possible. 
The Tractatus de latitudinibus formarum* 1 has been generally 

77 See Thorndike, History of Magic, III, 375. 

78 Duhem (fLtudes sur Leonard de Vinci, III, 478-79) would see in the Liber calcula- 
tionum the influence of Oresme and would characterize the work of Suiseth as "l'ceuvre 
d'une science senile et qui commence a radoter." Wieleitner ("Zur Geschichte der unend- 
lichen Reihen im christlichen Mittelalter," pp. 166-67) refers to Duhem in saying that 
Calculator rediscovered some of Oresme's examples. Thorndike has clearly shown, how- 
ever, that Oresme's work was subsequent to that of Calculator. 

79 Stamm ("Tractatus de continuo," p. 23), apparently referring to an edition which 
appeared in 1520 and which he mistakenly holds to be the first, asserts that Suiseth illus- 
trated his representation with geometrical figures. Such illustrations may well have been 
later interpolations, inasmuch as none such appear in the first edition (1477). Furthermore, 
French copyists sometimes drew diagrams on the edge of manuscripts of such Oxford 
works (Duhem, Etudes sur Leonard de Vinci, III, 449). 

80 "Quia formarum latitudines multipliciter variantur que multiplicitas difficiliter dis- 
cernitur nisi ad figuras geometricas consideratio referatur." Oresme, Tractatus de lati- 
tudinibus formarum, fol. 201 r . I have made use of a photostat of a manuscript of this work 
in the Bayerische Staatsbibliothek, Miinchen, cod. lat. 26889, fol. 201''— fol. 206'. See 
also Funkhouser, "Historical Development of the Graphical Representation of Statistical 
Data," p. 275; cf. also Stamm, op. cit., p. 24. 

81 See Wieleitner, "Der 'Tractatus'," for a comprehensive description of this work. For 
comments on the work of Oresme and a study of this period as a whole, see Duhem, 
fctudes sur Leonard de Vinci, III, 346-405. 



Medieval Contributions 81 

ascribed to Oresme and cited as his chief work on the subject. Although 
the views expressed in it are probably to be attributed to him, the 
treatise appears to be only a poor imitation, probably by a student, 82 
of a larger work by Oresme entitled Tractatus de figuratione potentiarum 
et mensurarum.™ This latter work, one of the fullest on the latitude of 
forms, was written probably before 1361. 84 It opens with the repre- 
sentation of variation by means of geometry, rather than with a dia- 
lectical exposition in terms of number, such as Calculator had given. 
Following the Greek tradition, which regarded number as discrete and 
geometrical magnitude as continuous, Oresme was led naturally to 
associate continuous change with a geometrical diagram. An intension, 
or rate of change by which a form acquires a quality, he imagined as 
represented by a straight line drawn perpendicular to a second line, 
the points of which represent the divisions of the time or space inter- 
val involved. 85 For example, a horizontal line or longitude may rep- 
resent the time or duration of a given velocity, and the vertical 
height or latitude the intensity of the velocity. 86 It will be seen that 
the terms "longitude" and "latitude" are here used to represent, 
in a general sense, what we should now designate as the abscissa and 
the ordinate. The work of Oresme does not, of course, represent the 
earliest use of a coordinate system, for ancient Greek geography had 
employed this freely; nor can his graphical representation be regarded 
as equivalent to our analytic geometry, 87 for it lacks the fundamental 

« Duhem, Etudes sur Uonard de Vinci, III, 399-400. 

83 This work is known also by other titles, such as De uniformitale et difformitate inien- 
tionum, and De conjlguratione qualitatum. See Wieleitner, "Ueber den Funktionsbegriff," 
for an extensive description of the portions of this work which are of significance in mathe- 
matics. For an analysis of those sections of the De configuratione qualitatum which deal 
with magic, see Thorndike, History of Magic, Vol. Ill, Chap. XXVI. 

84 Wieleitner, "Ueber den Funktionsbegriff," p. 198; Duhem, Etudes sur Uonard de 
Vinci, III, 375. 

85 "Omnis res mensurabilis extra numeros ymaginatur ad modum quantitates continue. 
Ideo opportet pro ejus mensuratione ymaginare puncta, superficies et lineas aut istorum 
proprietatis, in quibus, ut voluit Aristoteles, mensura seu proportio perprius reperitur .... 
Omnis igitur intensio successive acquisibilis ymaginanda est per rectam perpendiculariter 
erectam super aliquod puctum aut aliquot puncta extensibilis spacii vel subjecti." Wieleit- 
ner, "Ueber den Funktionsbegriff," p. 200. 

86 "Tempus itaque sive duratio erit ipsius velocitatis longitudo et ejusdem velocitatis 
intensio est sua latitude" Ibid., pp. 225-26. See also the diagrams in Tractatus de lati- 
tudinibus formarum, fols. 202 r and 205 f . 

87 Duhem has freely referred to Oresme as the inventor of analytic geometry. See, for 
example, his article on Oresme in the Catholic Encyclopedia; cf . also his Etudes sur Uonard 
de Vinci, HI, 386. 



82 Medieval Contributions 

notion that any geometric curve can be associated, through a co- 
ordinate system, with an algebraic equation, and conversely. 88 

However, Oresme's work marks a notable advance in mathematical 
analysis in that it associated the study of variation with the repre- 
sentation by coordinates. Although Aristotle had denied the existence 
of an instantaneous velocity, the notion had continued to be invoked 
implicitly, upon occasion, by Greek geometers and by Scholastic 
philosophers. Oresme, however, was apparently the first to take the 
significant step of representing an instantaneous rate of change by a 
straight line. 89 He could not, of course, give a satisfactory definition 
of instantaneous velocity, but he strove to clarify this idea by remark- 
ing that the greater this velocity is, the greater would be the distance 
covered if the motion were to continue uniformly at this rate. 90 
Maclaurin, in attempting to clarify the Newtonian idea of a fluxion, 
expressed himself very similarly almost four hundred years later, but 
a rigorous and clear definition could be given only after the concept 
of the derivative had been developed. Oresme, furthermore, was 
confused by the perplexing problem of the indivisible and the con- 
tinuum. In spite of his clear assertion that an instantaneous velocity 
is to be represented by a straight line, he accepted the dictum of 
Aristotle that every velocity persists throughout a time. 91 This view 
implies a form of mathematical atomism which, although underlying 
much of the thought leading to the calculus of Newton and Leibniz, 
has been rejected by modern mathematics. 

There is a widespread belief that the science of dynamics, which in 
the seventeenth century played such a significant r61e in the formation 
of the calculus, was almost entirely the product of the genius of 
Galileo, who "had to create ... for us" 92 the "entirely new notion 
... of acceleration." 93 That such a view is a gross misconception 
will be clear to anyone who makes even a cursory examination of the 
fourteenth-century doctrine of the latitude of forms. Oresme, for 

88 Cf. Coolidge, "The Origin of Analytic Geometry," p. 233. 

89 "Sed punctualis velocitas instantanea est ymaginanda per lineam rectam." Wieleit- 
ner, "Ueber den Funktionsbegriff," p. 226. 

90 "Verbi gratia; gradus velocitatis descencus est major, quo subjectum mobile magis 
descendit vel descenderet si continuaretur simpliciter." Ibid., p. 224. 

91 "Omnis velocitas tempore durat." Ibid., p. 225. 

92 Mach, The Science of Mechanics, p. 133. See also Hogben, Science for the Citizen^ 
p. 241. 

98 Mach, op. cit., p. 145. 



Medieval Contributions 



83 



example, had a clear conception not only of acceleration in general 
but also of uniform acceleration in particular. This is evident from 
his statement that if the velocitatio (acceleration) is uniformis, then 
the velocitas (velocity) is uniformiter dijformis; but if the velocitatio is 
dijformis, then the velocitas is dijformiter dijformis. Oresme went 
further and applied his idea of uniform rate of change and of graphical 
representation to the proposition that the distance traversed by a 
body starting from rest and moving with uniform acceleration is the 
same as that which the body would traverse if it were to move for 
the same interval of time with a uniform velocity which is one-half 
the final velocity. Later, in the seventeenth century, this proposition 
played a central r61e in the development both of infinitesimal methods 
and of Galilean dynamics. It had been stated earlier and in a more 
C 




D 
FIGURE 6 

general form by Calculator, but Oresme and Galileo gave to it a geo- 
metrical demonstration which, although not rigorous in the modern 
sense, was the best which could be furnished before the integral 
calculus had been established. 94 

The proof of the theorem is based upon the fact that the motion 
under uniform velocity, inasmuch as the latitude is the same at all 
times, is represented by a rectangle such as ABGF (fig. 6) and that 
the uniformly accelerated motion, in which the ratio of the change 
in latitude to the change in longitude is constant, 95 corresponds to 

94 In the mistaken belief that the work of Oresme preceded that of Calculator, Duhem 
(Ittudes sur Leonard de Vinci, III, 388-98) has called this proposition the law of Oresme. 

96 See Wieleitner, "Ueber den Funktionsbegriff," pp. 209-10; cf. also Traciatus de lati- 
ttidinibus formarum, fol. 204 r . Duhem, upon the basis of these observations, has unwar- 
rantedly asserted that Oresme "gives the equation of the right line, and thus forestalls 
Descartes in the invention of analytical geometry." See his article, "Oresme," in the 
Catholic Encyclopedia. 



84 Medieval Contributions 

the right triangle ABC. Oresme did not explicitly state the fact — 
this is, of course, demonstrated in the integral calculus — that the areas 
ABGF and ABC represent in each case the distance covered; but 
this seems to have been his interpretation, 96 inasmuch as from the 
congruence of the triangles CFE and EBG he concluded the equality 
of the distances. 97 This is perhaps the first time that the area under 
a curve was regarded as representing a physical quantity, but such 
interpretations were to become before long commonplaces in the 
application of the calculus to scientific problems. Oresme did not ex- 
plain why the area under a velocity-time curve represented the 
distance covered. It is probable, however, that he thought of the 
area as made up of a large number of vertical lines or indivisibles, 
each of which represented a velocity which continued for a very short 
time. Such an atomistic interpretation is in harmony with the views 
he expressed on instantaneous velocities and with the Scholastic 
interest in the infinitesimal. Several centuries later this interpretation 
was enunciated more boldly and vividly by Galileo, at a time when 
atomistic conceptions enjoyed an even greater popularity in both 
science and mathematics. 

Arguments similar to those given by Oresme and Calculator ap- 
peared also in the work of other men of the time, particularly at Oxford 
and Paris. 98 William of Hentisbery, a famous logician at Oxford who 
perhaps outdid Calculator in propounding sophisms on the subject of 
motion, clearly stated the law for uniformly difform variation. 99 
Marsilius of Inghen, at Paris, expounded this same law on the basis 
of Oresme 's geometrical representation. 100 The traditions developed 
at Oxford and Paris were continued also in the fifteenth century in 
Italy, where Blasius of Parma (or Biagio Pelicani), reputed the most 
versatile philosopher and mathematician of his time, explained the 
law similarly, in his Questiones super tractatum de latitudinibus for- 
marum. 101 This same principle for the uniformly difform acquisition 
of qualities was known at Paris in the sixteenth century to Alvarus 

96 Duhem, Etudes sur Leonard de Vinci, III, 394. 

97 Wieleitner, "Ueber den Functionsbegriff," p. 230. 

98 See Duhem, Etudes sur Leonard de Vinci, III, 405-81. 

99 Wieleitner ("Der 'Tractatus'," pp. 130-32, n.) says that Hentisbury's arguments are 
accompanied by diagrams, but Duhem (Eludes sur Leonard de Vinci, III, 449) says there 
is no geometrical demonstration given with his writings. 

h» Duhem, Etudes sur Leonard de Vinci, III, 399-405. 
101 See Amodeo, "Appunti su Biagio Pelicani da Parma." 



Medieval Contributions 85 

Thomas, 102 John Major, Dominic Soto, and others. 103 The principles 
of uniformly accelerated motion thus seem to have been common 
knowledge to the Scholastics from the fourteenth century to the 
sixteenth, and it is very probable that Galileo was familiar with their 
work and made use of it in his development of dynamics. 104 At all 
events, when the famous Discorsi of Galileo appeared in 1638, it 
contained a diagram and a type of argument resembling strikingly 
that previously given by Oresme. 105 

The efforts toward the mathematical representation of variation 
which resulted, in the fourteenth century, in the flurry of treatises on 
the latitude of forms, were tied up in many ways with other ques- 
tions related to the calculus. While he was discussing, in connection 
with latitudo difformiter difformis, a form represented graphically by a 
semicircle, the author of the Tractatus de latitudinibus formarum re- 
marked that the rate of change of an intensity is least at the point 
corresponding to the maximum intensity. 106 Great things have been 
read into this casual remark, for historians of mathematics have 
ascribed to its author the sentiments that the increment in the 
ordinate of the curve is zero at its maximum point, and that the 
differential coefficient vanishes at this point! 107 Ascribing such ideas 
to the author of the Tractatus is obviously unwarranted, inasmuch as 
they presuppose the clear conceptions of limit and differential quo- 
tient which were not developed until many centuries later. 108 Further- 

102 Wieleitner, "Zur Geschichte der unendlichen Reihen im christlichen Mittelalter," 
p. 154. 

103 Duhem, Etudes sur Leonard de Vinci, III, 531 ff. 

104 Zeuthen, Geschichte der Mathematik im XVI. und XVII. Jakrhundert, pp. 243-44. 

105 Mach (The Science of Mechanics, p. 131), probably unaware of the earlier work, has 
ascribed the diagram and ideas to Galileo. 

106 "In qualibet talis figura sua intensio terminatur ad summum gradum tarditatis et 
sua remissio incipit a summo gradu tarditatis ut in medio puncto aliqualis ubi terminatur 
intensio et incipit remissio, patet in figuris .c.d. et .d.c." Tractatus de latitudinibus forma- 
rum, fol. 205 f . The figures referred to (fol. 204") are much like semicircles. 

107 Curtze ("Ueber die Handschrift R. 4°, 2," p. 97) says that Oresme noticed in general 
that "der Zuwachs .... der Ordinate einer Kurve in der unmittelbaren Nahe eines Maxi- 
mums oder minimums gleich Null ist," and Moritz Cantor makes an even stronger state- 
ment when he says (Vorlesungen, II, 120) "Oresme's Augen offenbarte sich die Wahrheit 
des Satzes, den man 300 Jahre spater in die Worte kleidete, an den Hohen- undTiefpunkten 
einer Curve sei der Differentialquotient der Ordinate nach der Abscisse Null." 

108 Wieleitner ("Der 'Tractatus,' " p. 142) says "Ich halte aber die Bemerkung doch nur 
fur rein anschauungsmassig"; and Timtchenko ("Sur un point du 'Tractatus de lati- 
tudinibus formarum' de Nicholas Oresme") agrees, saying "qu'il 6tait encore assez loin du 

theoreme exprimd par la formule — = (pour le sommet de la figure)." 

dx 



86 Medieval Contributions 

more, the writer had apparently no idea of generalizing the statement 
by extending the conclusion to other cases. His remarks do show 
clearly, however, how fruitful the idea of the latitude of forms was to 
be when it entered the geometry and algebra of later centuries, to 
become eventually the basis of the calculus. The Scholastic philoso- 
phers were striving to express their ideas in words and geometrical 
diagrams, and were not so successful as we who realize, and can 
make use of, the economy of thought which mathematical notation 
affords. 

Further consideration by Oresme of a latUudo dijjormiter difformis 
led him, as it had also Suiseth somewhat earlier, to another topic 
essential in the development of the calculus — that of infinite series. 
In the same manner as Calculator, he considered a body moving 
with uniform velocity for half a certain period of time, with double 
this velocity for the next quarter of the time, with three times this 
velocity for the next eighth, and so ad infinitum. He found in this 
case that the total distance covered would be four times that covered 
in the first half of the time. However, whereas Calculator had had 
recourse to devious verbal argumentation in his justification of this 
result, the method of Oresme was here (as in his earlier work) geo- 
metrical and consisted in the comparison of the areas corresponding to 
the distances involved in the graphical representation of the motion. 109 

In this manner also he handled another similar case in which the 
time was divided into parts j, ts, -is, . . . the velocity increasing in 
arithmetic proportion as before. The total distance he found this time 
to be -^ that covered in the first subinterval. Oresme then went on 
to still more complicated cases. For example, he assumed that during 
the first half of the time interval the body moved with uniform 
velocity, then for' the next quarter of the time with motion uniformly 
accelerated until the velocity was double the original, then for the 
next eighth of the time uniformly with this final velocity, then for 
the next sixteenth with uniform acceleration until the velocity was 
again doubled, and so on. Oresme found that in this case the total 
distance (or qualitas) is to that of the first half of the time as seven 
is to two. 110 This is equivalent to summing geometrically to infinity 
the series i + f + i+A + i+A + Tnr + A + 

"• See Wieleitner, "Ueber den Funktionsbegriff," pp. 231-35. 

«" Wieleitner, Ibid., p. 235. 



Medieval Contributions 87 

Calculator and Oresme were not the only Scholastics who were 
concerned with such infinite series. In the anonymous tract written 
before 1390, A est unum calidum, nx some of their results on series were 
rediscovered. More than two centuries later, in 1509, similar work 
appeared in the Liber de triplici motu of Alvarus Thomas, at Paris. 
This book was intended to serve as an introduction to the Liber 
calculationum of Suiseth and in it are handled several of the series 
given earlier by Calculator and Oresme. However, the author went on 
to similar but more complicated cases, in which he found (in the 
manner of Calculator and Oresme) the sums to infinity of the series 

t. T 4 2^8 2 2 ^ 16 2» ^ ' ' * ^3 2^6 2 a + 12 2* 

-+- . . . , these sums being f and ^ respectively. 112 Alvarus Thomas 
remarked that innumerable other such series can be found. In some 
of these the sums involve logarithms, so that he could not arrive at 
the sum exactly, but gave it approximately between certain integers. 

Thus he said that the sum of 1 + -j • 5 + jj * 2 1 ~*~ 3 ' 2 1 "*~ ' * * ( wn i cn 
results in 2 -f log 2) lies between 2 and 4. 113 

It must be remembered that these infinite series were not handled 
by the Scholastic philosophers as they are now in the calculus, for 
they were given rhetorically, rather than by means of symbols, and 
were bound up with the concept of the latitude of forms. Furthermore, 
the results were found either through verbal arguments or geo- 
metrically from the graphical representation of the form, rather than 
by means of arithmetic considerations based upon the limit concept. 
Had their work been more closely associated with the geometrical 
procedures of Archimedes and less bound up with the philosophy of 
Aristotle, it might have been more fruitful. It could then have antic- 
ipated Stevin and the geometers of the seventeenth century in the 
elimination, from the classical method of exhaustion, of the argu- 
ment by a reductio ad absurdum, and in the substitution of an 
analysis suggestive of the limit idea. 

The fourteenth-century work at Paris and Oxford on the latitude 

m Wieleitner, "Zur Geschichte der unendlichen Reihen," p. 167; cf. also Duhem, 
tiudes sur Leonard de Vinci, III, 474-77. 

m Wieleitner, "Zur Geschichte der unendlichen Reihen"; Duhem, Etudes sur Leonard 
de Vinci, HI, 531 ff. 

m Wieleitner, "Zur Geschichte der unendlichen Reihen," pp. 161-64; see also Duhem, 
Ittudes sur Leonard de Vinci, HI, 540-41. 



88 Medieval Contributions 

of forms and related topics was not forgotten during the subsequent 
decline of Scholasticism, for the doctrines spread throughout the 
Italian universities. 114 In particular, they were taught at Pavia, 
Bologna, and Padua by Blasius of Parma. 115 He, in turn, was referred 
to a century later by Luca Pacioli and Leonardo da Vinci; and Nich- 
olas of Cusa was very likely influenced by him. 116 The work of Calcu- 
lator and Oresme was much admired by the men of the fifteenth and 
sixteenth centuries, several printed editions of both the Liber cal- 
culationum and the De latitudinibus formarum, as well as commen- 
taries on each, appearing in the interval from 1477 to 1520. 117 In the 
De subtilitate of Cardan, Calculator is classed among the great, along 
with Archimedes, Aristotle, and Euclid. 118 Galileo, we have said, gave 
in his dynamics a geometrical demonstration almost identical with 
that of Oresme, and in another place referred to Calculator and 
Hentisbery on activity in resisting media. 119 Descartes likewise, in 
his attempt to determine the laws of falling bodies, employed argu- 
ments closely resembling those of Oresme. 120 As late as the end of the 
seventeenth century the reputation of Calculator was such that 
Leibniz on several occasions referred to him as almost the first to 
apply mathematics to physics and as one who introduced mathe- 
matics into philosophy. 121 

In spite of the reputation maintained by the great proponents of 
the doctrine of the latitude of forms, however, the type of work they 
represented was not destined to be the basis of the decisive influence 
in the development of the methods of the calculus. The guiding 

114 Duhem, Etudes sur Lionard de Vinci, III, 481-510. 

115 See Thorndike, History of Magic, IV, 65-79. 

116 See Amodeo, "Appunti su Biagio Pelicani da Parma," III, 540-53. 

117 The Liber calculationum was published at Padua in 1477 and 1498, and again at 
Venice in 1520; De latitudinibus formarum appeared at Padua in 1482 and 1486, at Venice 
in 1505, and at Vienne in 1515. 

™ Cardan, Opera, III, 607-8. 

lu "Secunda dubitatio: quomodo se habent primae qualitates in activitate et resistentia. 
De hac re lege Calculatorem in tractatu De reactione, Hentisberum in sophismate." See 
Opere, I, 172. 

™CEuvres,X, 58-61. 

m "Quis refert, primum prope eorum, qui mathesin ad physicam applicarunt, fuisse 
Johan. [sic] Suisset calculatorem ideo, scholasticis appellatum": Letter to Theophilus 
Spizelius, April 7/17, 1670. Opera omnia (Dutens), V, 347. "Vellem etiam edi scripta 
Suisseti, vulgo dicti calculatoris, qui Mathesin in Philosophicam Scholasticam introduxit, 
sed ejus scripta in Cottonianis non reperio." Letter to Thomas Smith, 1696. Opera omnia 
(Dutens), V, 567. See also Opera Omnia (Dutens), V, 421. 



Medieval Contributions 89 

principles were to be supplied by the geometry of Archimedes, al- 
though these were to be modified by kinematic notions derived from 
the quasi-Peripatetic disputations of the Scholastic philosophers on 
the subject of variation. As early as the beginning of the fifteenth 
century, the mensurational science and mathematics of Archimedes 
were becoming more influential, so that men like Blasius of Parma in 
some instances preferred his explanations to those of Aristotle. Blasius 
of Parma is said to have written on infinitesimals 122 as well as upon 
other mathematical topics, but no such works are known to be extant. 
We do, however, have the work of Cardinal Nicholas of Cusa, 
which illustrates the influence both of the Scholastic speculations 
(perhaps through Blasius) and the work of Archimedes, as well as 
another tendency which grew up, particularly on German soil, in the 
fifteenth century. With the decline of the extremely rational and 
rigorous thought of Scholasticism (the overpreciseness of which was 
anathema to the growing Humanism of the time), 123 there was a trend 
toward Platonic and Pythagorean mysticism. 124 This may well have 
been in some measure responsible for the prevalence of mysticism 126 
at the time. On the other hand, the development of science in Italy, 
culminating in the dynamics of Galileo, owed much to the mathe- 
matical philosophy of Plato and Pythagoras. 126 From the point of 
view of the rise of the calculus, this pervasive Platonism exerted for 
several centuries a not altogether unfortunate influence upon mathe- 
matics, for it allowed to geometry what Aristotle's philosophy and 
Greek mathematical rigor had denied — the free use of the concepts 
of infinity and the infinitesimal which Platonic and Scholastic phi- 
losophers had fostered. Nicholas of Cusa was not a trained mathe- 
matician, but he was certainly well acquainted with Euclid's Elements 
and had read Archimedes' works. Although he was primarily a theo- 
logian, mathematics nevertheless constituted for him, as for Plato, 

m Hoppe ("Geschichte der Infinitesimalrechnung," p. 179) says "Er ist also in dieser 
Beziehung ein Vorlaufer von Cavalieri, von dem ihn 148 Jahre trennen." Blasius died in 
1416, but probably Hoppe has in mind the 1486 edition of Blasius' Questiones, which is 
separated from Cavalieri's Geometria in&ivisibilibus of 1635 by about 148 years. 

123 Thorndike, History of Magic, III, 370. 

m Lasswitz, Geschichte der Atomistik, I, 264-65; see also Burtt, The Metaphysical Founda- 
tions of Modern Physical Science, pp. 18-42. 

m Cf. Thorndike, Science and Thought in the Fifteenth Century, p. 22. 

126 Strong, Procedures and Metaphysics, pp. 3-4; see also Cohn, Geschichte des Unend- 
lichkeitsproblems, p. 95. 



90 Medieval Contributions 

the basis of his whole system of philosophy. 127 For him mathematics 
was not restricted, as for Aristotle, to the science of quantity, but was 
the necessary form for an interpretation of the universe. 128 Aristotle 
had insisted on the operational character of mathematics and had 
rejected the metaphysical significance of number, but Cusa revived 
the Platonic arithmology. 129 He again associated the entities of mathe- 
matics with ontological reality and restored to mathematics the cos- 
mological status which Pythagoras had bestowed upon it. Further- 
more, the validity of its propositions was regarded by him as established 
by the intellect, with the result that the subject was not bound by the 
results of empirical investigation. 

This view of mathematics as prior to, or at least independent of, 
the evidence of the senses encouraged speculation and allowed the 
indivisible and the infinite to enter, so long as no inconsistency in 
thought resulted. Such an attitude enriched the subject and eventu- 
ated in the methods of the calculus, but this advantage was gained at 
the expense of the rigor which characterized ancient geometry. The 
logical perfection in Euclid found no counterpart in modern mathe- 
matics until the nineteenth century. From the point of view of the 
development of the calculus, this was not an unfortunate situation, 
for the infinite and the infinitely small were allowed almost free reign, 
even though it was about four hundred years before a logically satis- 
factory basis for these notions was found and the limit concept was 
definitely and rigorously established in mathematics. 

In spite of Aristotle, the infinitely large and the infinitely small had 
been employed somewhat surreptitiously by Archimedes in his geom- 
etry, and the Scholastics had freely discussed these notions in their 
dialectical philosophy; but with Nicholas of Cusa these ideas became, 
together with an unnecessarily large admixture of Pythagorean and 
theological mysticism, 130 a recognized part of the subject matter of 
mathematics. 

The definitions given by Nicholas of Cusa for the infinitely large 
(that which cannot be made greater), and for the infinitely small 

m Schanz, Der Cardinal Nicolaus von Cusa als Mothematiker, pp. 2, 7. 
128 Lob, Die Bedeutung der Mathematik fur die Erkenntnislehre des Nikolaus von Kues, 
pp. 36-37; cf. also Max Simon, Cusanus als Mothematiker. 
»» Cf. L. R. Heath, The Concept of Time, p. 83. 
180 Cohn, Geschichte des UnendlichkeitsproUems, p. 127. 



Medieval Contributions 91 

(that which cannot be made smaller) 131 are unsatisfactory, and his 
mathematical demonstrations are not always unimpeachable. His 
work is significant, however, in that it made use of the infinite and 
the infinitesimal, not merely as potentialities, as had Aristotle, but as 
actualities which are the upper and lower bounds of operations upon 
finite magnitudes. Just as the triangle and the circle were to Nicholas 
of Cusa the polygons with the smallest and the greatest number of 
sides, so also zero and infinity were to him the lower and upper bounds 
of the series of natural numbers. 132 His view was colored, moreover, 
by his favorite philosophical doctrine — derived, perhaps, from the 
Scholastic quasi-theological disputations on the infinite — that a finite 
intelligence can approach truth only asymptotically. Consequently, 
the infinite was the source and means, and at the same time the 
unattainable goal, of all knowledge. 133 Cusa was led by this attitude 
to view the infinite as a terminus ad quern to be approached only by 
going through the finite 134 — an idea which represents a striving' toward 
the limit concept 135 which was to be developed during the next five 
hundred years. 

Upon the basis of this view, Nicholas of Cusa proposed a char- 
acteristic quadrature of the circle. If, contrary to Greek ideas of 
congruence, the circle belongs among the polygons as the one with 
an infinite number of sides and with apothem equal to the radius, its 
area may be found by the same means as that employed for any 
other polygon: by dividing it up into a number (in this case an infinite 
number) of triangles, 136 and computing the area as half the product 
of the apothem and the perimeter. Nicholas of Cusa added to this 
explanation of the measurement of the circle an Archimedean proof, 
using inscribed and circumscribed polygons, and a proof by the 
reductio ad absurdum; but when his method was used later by Stevin, 
Kepler, and others, the Archimedean proof was abandoned, an ele- 
mentary equivalent of the limit concept being considered sufficient. 

As Plato had opposed the Democritean atomic doctrine and yet 
attempted to link the continuum with indivisibles, so also Cusa ob- 
jected to Epicurean atomism, but at the same time regarded the line 

131 Lorenz, Das Unendliche bet Nicolaus von Cues, p. 36. u * Lorenz, op. cit., p. 2. 

133 Cohn, op. cit., p. 87. m Lasswitz, Geschichteder Atomistik, I, 282. 

135 Cf. Simon, "Zur Geschichte und Philosophie der Differentialrechnung," p. 116. 

136 Schanz, Nicolaus von Cusa, pp. 14r-15. 



92 Medieval Contributions 

as the unfolding of a point. In a similar manner, also, he held that 
while continuous motion is thinkable, in actuality it is impossible, 137 
inasmuch as motion is to be regarded as composed of serially ordered 
states of rest. 138 These views call to mind the modern mathematical 
continuum and the so-called static theory of the variable, but to 
Nicholas of Cusa must not be ascribed any of the precision of state- 
ment which now characterizes these matters. He did not make clear 
in what manner the transition from the continuous to the discrete 
is made. He asserted that although in thought the division of con- 
tinuous magnitudes, such as space and time, may be continued indefi- 
nitely, nevertheless in actuality this process of subdivision is limited 
by the smallness of the parts obtained — that is, of the atoms and 
instants. 139 

The looseness of expression found toward the middle of the fifteenth 
century in the formulation by Nicholas of Cusa of his views as to the 
nature of the infinitesimal and the infinitely large was paralleled about 
two centuries later by a similar lack of clarity in the multifarious 
methods of indivisibles to which they to some extent gave rise. These 
latter procedures, in turn, led to the differential and the integral cal- 
culus. It is not to be concluded, however, that the views of Cusa are 
to be construed as indicating any renaissance in mathematics, nor as 
heralding the rise of the new analysis. The cardinal, absorbed as he 
was in matters concerning church and state, did not contribute any 
work of lasting importance in mathematics. Regiomontanus was per- 
fectly justified in his criticism 140 of the repeated attempts which were 
made by Cusa to square the circle. Nevertheless, as Plato's thought 
may have been instrumental in the use of infinitesimals made by 
Archimedes in his investigations preliminary to the application of the 
rigorous method of exhaustion, so also the speculations of Nicholas 
of Cusa may well have induced mathematicians of a later age to em- 
ploy the notion of the infinite in conjunction with the Archimedean 
demonstrations. 

Leonardo da Vinci, who was strongly influenced, through Nicholas 

137 L. R. Heath, The Concept of Time, p. 82. 

138 Cohn, Geschichte des Unendlichkeits problems, p. 94. 

13 » L. R. Heath, The Concept of Time, p. 82; Cohn, op. cit., p. 94; Lasswitz, Geschichte der 
Atomistik, II, 276 ff. 
140 See Kaestner, Geschichte der Matltematik, I, 572-76. 



Medieval Contributions 93 

of Cusa, by Scholastic thought, 141 and who was acquainted also with 
the work of Archimedes, is said to have employed infinitesimal consid- 
erations in finding the center of gravity of a tetrahedron by thinking 
of it as made up of an infinite number of planes. We cannot, how- 
ever, be sure of his point of view. 142 Cusa's ideas were more clearly 
expressed in the work of Michael Stifel, who held that a circle may 
correctly be described as a polygon with an infinity of sides, and that 
before the mathematical circle there are all the polygons with a 
finite number of sides, just as preceding an infinite number there are 
all the given numbers. 143 Somewhat later Francois Viete also spoke 
of a circle as a polygon with an infinite number of sides, 144 thus show- 
ing that such conceptions were widely held in the sixteenth century. 
The fullest expression of Nicholas of Cusa's mathematical thoughts 
on the infinite and the infinitesimal, however, are found in the work 
of Johann Kepler, who was strongly influenced by the cardinal's 
ideas — speaking of him as divinus mihi Cusanus — and who was like- 
wise deeply imbued with Platonic and Pythagorean mysticism. It 
was probably the imaginative use by Cusa of the concept of infinity 
which led Kepler to his principle of continuity — which included 
normal and limiting forms of a figure under one definition, and in 
accordance with which the conic sections were regarded as con- 
stituting a single family of curves. 145 Such conceptions often led to 
paradoxical results, inasmuch as the notion of infinity had not yet 
received a sound mathematical basis; but Kepler's bold views suggested 
new paths which were be to very fruitful later. Kepler lived a century 
and a half after the time of Cusa. Moreover, unlike his predecessor, 
although his early training was in theology, he was primarily a mathe- 
matician. He was thus able, to the fullest extent, to take advantage 

141 Duhem, Etudes sur Leonard de Vinci, II, 99 ff . 

142 Libri, Histoire des sciences mathematiques en Ilalie, III, 41 ; Duhem, Etudes sur Lionard 
de Vinci, I, 35-36. The former asserts that Leonardo used indivisibles; the latter questions 
this. Duhem is wrong when he says in this connection (cf . Les Origines de la statique, II, 
74) that Archimedes had restricted himself to plane figures in his work on the center of 
gravity, since in the Method the centers of gravity of segments of a sphere and of a parabo- 
loid are determined. 

143 Cf. Gerhardt, Gesckichte der Mathematik in Deutschland, pp. 60-74. 

144 Moritz Cantor, Vorlesungen, II, 539-40. 

145 This is indicated by his statement, ". . . dico, lineam rectam esse hyperbolarum 
obtussissimam. Et Cusanus infinitum circulum dixit esse lineam rectam. . . . Plurima 
talia sunt, quae analogia sic vult efferi, non aliter." Kepler, Opera omnia, II, 595. 



94 Medieval Contributions 

of the accessibility of Archimedes' works resulting from their transla- 
tion and publication during the middle of the sixteenth century. 

It has been remarked that the medieval period added little to the 
classical Greek works in geometry or to the theory of algebra. Its 
contributions were chiefly in the form of speculations, largely from 
the philosophical point of view, on the infinite, the infinitesimal, and 
continuity, as well as of new points of view with reference to the 
study of motion and variability. Such disquisitions were to play a 
not insignificant part in the development of the methods and con- 
cepts of the calculus, for they were to lead the early founders of the 
subject to associate with the static geometry of the Greeks the graph- 
ical representation of variables and the idea of functionality. 

Both Newton and Leibniz, as well as many of their predecessors, 
sought for the basis of the calculus in the generation of magnitudes — 
a point of view which may be regarded as the most notable contribu- 
tion of Scholastic philosophy to the development of the subject. 
However, the sound mathematical basis for the seventeenth-century 
elaboration of the infinitesimal procedures into a new analysis was 
supplied by the mensurational treatises of Archimedes which had been 
composed much earlier. These had become scattered and some had 
been lost, although a number of them were familiar to the Arabs, and 
although Archimedean manuscripts were known in the Scholastic 
period. 146 No significant additions to his results were made until after 
the appearance of printed editions of his treatises. In 1543 the Italian 
mathematician, Nicolo Tartaglia, published at Venice portions of 
Archimedes' work, including that on centers of gravity, the quadrature 
of the parabola, the measurement of the circle, and the first book 
on floating bodies. The editio princeps of Venatorius appeared at Basel 
in 1544, and in 1558 the important translation by Federigo Com- 
mandino appeared at Venice. Commandino himself composed a 
treatise in the Archimedean manner — Liber de centro gravitatis soli- 
dorum — showing that the study of the methods of the great Syracusan 
mathematician had advanced to the point where new contributions 
could be made. Shortly after this, however, numerous attempts were 
made by Kepler and others to replace the almost tediously rigorous 

148 See Thorndike and Kibre, Catalogue of Incipits, for citations of some dozen manu- 
scripts of the twelfth, thirteenth, fourteenth, and fifteenth centuries. 



Medieval Contributions 95 

arguments of Archimedes by new methods which should be equivalent 
to the older ones but at the same time possess a simplicity and an 
ease of application to new problems which was lacking in the method 
of exhaustion. 

It is precisely the search for such substitutes which was to lead 
within the next century, with the help of the Scholastic speculations, 
to the methods of the calculus. For the sake of convenience and 
unity, we shall make these anticipations the subject of the following 
chapter. 



IV. A Century of Anticipation 

THE DEVELOPMENT of the concepts of the calculus may be 
considered to have begun with the Pythagorean effort to com- 
pare — through the superposition of geometrical magnitudes — lengths, 
areas, and volumes, in the hope of thus associating with each configura- 
tion a number. Thwarted in this by the problem of the incommensur- 
ability of such magnitudes, it was left for later Greek geometers to 
circumvent such direct comparison through the method of exhaustion 
of Eudoxus. Through this method the need for the infinite and the 
infinitesimal had been obviated, although such notions had been con- 
sidered by Archimedes as suggestive heuristic devices, to be used in 
the investigation of problems concerning areas and volumes which 
were preliminary to the intuitively clear and logically rigorous proofs 
given in the classical geometrical method of exhaustion. 

During the later Middle Ages the conceptions of the infinite and 
the infinitesimal, and the related ideas of variation and the continuum, 
were more freely discussed and utilized. Because, however, the in- 
terest in such s peculations remained less closely associated with geom- 
etry than with metaphysics or science, or even with attempts to explain 
the possibility of natural magic, 1 the primary influence leading to the 
calculus was n^t supplied by the doctrines of the Scholastic philoso- 
phers but by 1he greater enthusiasm for the methods of Archimedes 
which was mar if est in the late sixteenth century and which continued 
throughout the whole of the century following. Whereas a multiplicity 
of printed editi :>ns of the works of Jordanus Nemorarius, Bradwardine, 
Calculator, Orcsme, Hentisbery, and other medieval scholars had ap- 
peared in the ]ate fifteenth and early sixteenth centuries, there arose 
toward the mic die of the latter century a strong opposition, illustrated 
by the attitude of Ramus, to Aristotelianism and Scholastic method- 
ology. 2 It was during the height of this reaction that the works of 
Archimedes ap reared in numerous editions and, admiring Archimedes, 

1 Thorndike, Hi %tory of Magic, III, 371. 

2 Johnson and Larkey, "Robert Recorde's Mathematical Teaching and the Anti- 
Aristotelian Mover lent." 



A Century of Anticipation 97 

the men of the time refused to recognize the work of the Middle 
Ages. 3 

In the mathematics of this period, nevertheless, the intrusion of in- 
fluences from the Scholastic age is easily discernible in the attempts to 
reconcile with the thought of Archimedes the newly discovered infin- 
itesimal methods which were fostered. There is noticeable in this de- 
velopment, as well, a tendency foreign alike to Archimedes and to the 
Scholastic period — a deeper interest in the Arabic algebra which had 
been developing in Italy, in which the concept of infinity did not 
figure. The algebra of Luca Pacioli's Summa de arithmetica was not 
greatly advanced over that to be found in Leonardo of Pisa almost 
three hundred years earlier, 4 but in the sixteenth century the subject 
was assiduously studied again. Before 1545 the cubic had been solved 
by Tartaglia and Cardan, and the quartic by Ferrari; and thereafter 
a freer use of irrational, negative, and imaginary numbers was made 
by Cardan, Bombelli, Stifel, and others. The Greeks had not regarded 
irrational ratios as numbers in the strict sense of the word, and the 
attitude in the medieval period had been similar. Bradwardine as- 
serted that an irrational proportion is not to be represented by any 
number; 5 and Oresme, in discussing the popular question of whether 
the celestial motions were commensurable or incommensurable, 
concluded that geometry favored the latter but arithmetic the 
former. 6 

The Hindus and the Arabs, on the other hand, had not clearly dis- 
tinguished between rational and irrational numbers. On adopting the 
Hindu-Arabic algebra, the sixteenth-century mathematicians had 
continued to employ the irrational ratio. They now recognized this as 
a number, but they stigmatized it, following Leonardo of Pisa, as a 
numerus surdus, and they continued to interpret it geometrically as a 
ratio of lines. 7 Negative quantities, admitted by the Hindus but not 
by the Greeks or the Arabs, were accepted in the sixteenth century as 
numeri falsi or ficti, but in the following century these were recog- 

3 Duhem, Les Origines de la statique, I, 212. 4 Cajori, History of Mathematics, p. 128 

5 Hoppe, "Zur Geschichte der Infinitesimalrechnung," p. 158. 

• Thorndike, History of Magic, III, 406. 

7 See Pringsheim, "Nombres irrationnels et notion de limite," pp. 137-40; Fink, A Brief 
History of Mathematics, pp. 100-1. However, Kepler, as late as 1615 spoke of the irrational 
as "ineffable." Opera omnia, IV, 565. 



98 A Century of Anticipation 

nized as numbers in the strict sense of the word. 8 Imaginary numbers 
also were regularly employed after the sixteenth century, although 
they continued to occupy an anomalous position in mathematics until 
the time of Gauss. 

Such generalizations of number, although not at the time based 
upon satisfactory definitions, were influential later in leading to the 
limit concept and to the arithmetization of mathematics. More im- 
portant than this, in the development of the algorithm of the calculus, 
was the systematic introduction, during the later sixteenth century, of 
symbols for the quantities involved in algebraic relations. 

As early as the thirteenth century, letters had been used as symbols 
for quantities by Jordanus Nemorarius in his science and mathematics. 
Their establishment as symbols of the abstract quantities entering into 
algebra, however, was largely the work of the great French mathe- 
matician Francois Viete, 9 who used consonants to represent known 
quantities and vowels for those unknown. He distinguished arith- 
metic, or logistica numerosa, from algebra, or logistica speciosa, thus 
making the latter a calculation with letters rather than with numbers 
alone. 

This literal symbolism was absolutely essential to the rapid progress 
of analytic geometry and the calculus in the following centuries, 10 for 
it permitted the concepts of variability and functionality to enter into 
algebraic thought. The improved notation led also to methods which 
were so much more facile in application than the cumbrous geometrical 
procedures of Archimedes, of which they were modifications, that 
these methods were eventually recognized as forming a new analysis 
— the calculus. The period during which this transformation took place 
may be considered as the century preceding the work of Newton and 
Leibniz. 11 

Numerous translations of the works of Archimedes had been made 
during the middle of the sixteenth century, and soon after this mathe- 

8 See Fine, Number System, p. 113; Paul Tannery, Notions historiques, pp.,333-34; Fehr, 
"Les Extensions de la notion de nombre dans leur developpement logique et historique." 

9 Cajori, History of Mathematics, p. 139. 

10 Cf. Karpinski, "Is There Progress in Mathematical Discovery?" p. 47. 

11 For a comprehensive account of the methods developed during this period, see 
Zeuthen, Geschichte der Mathcmatik itn -XVI. una 1 XVII. Jahrhundert. Zeuthen's account 
is an outline of the development of the subject in this interval, rather than of fundamental 
concepts, and so includes a large amount of mathematical detail. 



A Century of Anticipation 99 

maticians had reached the point at which original contributions could 
be made to the classical work of the Greeks. This is evidenced by the 
fact that Commandino in 1565 published a work of his own on centers 
of gravity. In this he proved, among other things, that the center of 
gravity of a segment of a paraboloid of revolution is situated on the 
axis two-thirds of the distance from the vertex to the base. 12 This was 
a proposition which Archimedes had, incidentally, demonstrated by 
infinitesimals in a treatise, the Method, which was apparently not 
known at that time. Commandino 's proof followed the orthodox style 
of the method of exhaustion. Such extensions, however, had perhaps 
less influence on the development of the calculus than did certain 
striking innovations in method introduced by succeeding mathe- 
maticians. Perhaps the first of such significant modifications is that 
advanced by Simon Stevin of Bruges in 1586, very nearly a century 
before the first printed work on the calculus by Leibniz, in 1684, and 
that of Newton in 1687. 

Stevin was essentially an engineer and a practical-minded scientist. 
For this reason he had perhaps less regard for the philosophy of science 
and the exigencies of mathematical rigor than for technological tradi- 
tion and methodology. 13 As a result, Stevin did not merely imitate, as 
had Commandino, Archimedes' use of the method of exhaustion: he 
accepted the direct portion of his characteristic proof as sufficient to 
establish the validity of any proposition that required it, without add- 
ing in every case the formal reductio ad absurdum required by Greek 
rigor. Furthermore, he frequently omitted, as we do in the integral 
calculus, one of the approximating figures which Archimedes had used, 
being satisfied with either the inscribed or the circumscribed figure 
only. Stevin demonstrated as follows (in his work on statics, in 1586) 
that the center of gravity of a triangle lies on its median. Inscribe in 
the triangle ABC a number of parallelograms of equal height, as il- 
lustrated (fig. 7). The center of gravity of the inscribed figure will lie 
on the median, by the principle that bilaterally symmetrical figures 
are in equilibrium (a principle used by Archimedes in proving the law 
of the lever, and also by Stevin in his well-known demonstration of 
the law of the inclined plane). However, we may inscribe in the tri- 

a Commandino, Liber de centro gravitatis solidorum, fol. 40"-41 B . 
u Strong, Procedures and Metaphysics, pp. 91-113. 



100 



A Century of Anticipation 



angle an infinite number of such parallelograms, for all of which the 
center of gravity will lie on AD. Moreover, the greater the number of 
parallelograms thus inscribed, the smaller will be the difference be- 
tween the inscribed figure and the triangle ABC. If, now, the "weights" 
of the triangles ABD and ACD are not equal, they will have a certain 
fixed difference. But there can be no such difference, inasmuch as each 
of these triangles can be made to differ by less than this from the sums 
of the parallelograms inscribed within them, which are equal. There- 
fore the "weights" of ABD and ACD are equal, and hence the center 
of gravity of the triangle ABC lies on the median AD. U 




Exactly analogous demonstrations were given by Stevin of propo- 
sitions on the centers of gravity of plane curvilinear figures, including 
the parabolic segment. These proofs given by Stevin indicate the direc- 
tion in which the method of limits was developed as a positive con- 
cept. The Greek method of exhaustion had not boldly concluded, as 
did Stevin, that inasmuch as the difference could, by continued sub- 
division, be shown to be less than any given quantity, there could as 
a result be no difference. The Greeks felt constrained in every case to 
add the full reductio ad absurdum proof to show the equality. Stevin 
did not, of course, speak of the triangle as the limit of the sum of the 

14 Stevin, Hypomnemata mathematica, IV, 57-58; cf . also Bosnians, "Le Calcul infinitesi- 
mal chez Simon Stevin." 



A Century of Anticipation 



101 



inscribed parallelograms; but it would require only slight changes in 
his method — largely in the nature of further arithmetization and the 
use of greater precision in terminology — to recognize in it our modern 
method of limits. 

That Stevin regarded his approach to these problems as a significant 
modification of the method of Archimedes may be indicated in his 
proof of the proposition on the center of gravity of a conoidal segment 
— a proposition "the demonstration of which the ingenious and subtle 
mathematician Fredericus Commandinus gives in proposition 29 de 
solidorum centrobaricis, and which is arranged as follows in accordance 




D 
FIGURE 8 

with our custom and method." 15 Circumscribe about the segment 
ABC two cylindrical segments FGBC and MLIK as illustrated (fig. 8). 
Now the centers of gravity of these cylindrical segments are, by the 
principle of the equilibrium of bilaterally symmetrical figures, at the mid- 
points, N and O, of their axes, AH and HD, respectively; and the 
center of gravity of the entire circumscribed figure is at R such that 
NR = 2RO. Letting E be the point such that AE = 2ED, it can be 
shown that ER = tsAD. If, now, one similarly circumscribes about 

15 Stevin, Hypomnemata mathematica, II, 75-76. 



102 A Century of Anticipation 

the segment ABC four such cylindrical segments of equal height, the 
center of gravity of this circumscribed figure is found to lie above E at 
a point L such that EL = -riAD. Successively doubling the number of 
these cylinders, the center of gravity of the circumscribed figure re- 
mains always above E and will differ from E by teAD, tsAD, and so 
on. Thus the center of gravity descends, approaching E more and more 
closely. Modern mathematics would now conclude that E is the limit 
of the center of gravity of the circumscribed figure and therefore is the 
center of gravity of the conoidal segment. Stevin, however, reached 
this conclusion only after cautiously observing that similarly the center 
of gravity of the analogous inscribed figure ascended toward E in the 
same manner. 

The demonstration given by Stevin, in the above proposition in 
terms of the sequence ts, ih, tV, sV, • . ., is comparable to that em- 
ployed by Archimedes in his quadrature of parabola, in which the 
series 1 + £ + T \ + ^ + . . . figured. Both the Greek mathemati- 
cian and the Flemish engineer, in their use of such sequences and 
series, stopped short of the limit concept. Neither thought of such a 
sequence or series as carried out to an infinite number of terms in the 
modern sense. Archimedes explicitly stopped with the nth term j^ 
and added the term, ^ • 7—^, representing the remainder of the series; 

Stevin used the word infinite in the Peripatetic sense of potentiality 
only — the sequence could be continued as far as desired, and the error 
consequently made as small as one pleased. For this reason Archimedes 
had been constrained to complete his work through the demonstration 
by a reductio ad absurdum; Stevin, although he adduced no such for- 
mal argument, had recourse also to supplementary demonstrations, 
such as the inclusion, in the above proposition on the conoid, of a 
second sequence approaching the point from the other side. This hesi- 
tation on the part of Stevin to accept as sufficient the notion of the 
limit of an infinite sequence is apparent also in a proposition in his hy- 
drostatics, which represented perhaps his nearest approach to the 
method of limits. Here he supplemented the "mathematical demon- 
stration" of propositions, carried out as above, by a "demonstration 
by numbers," suggested, perhaps, by the then recent Italian work in 
algebra and mensuration, and encouraged by the neglect of geometry 



A Century of Anticipation 103 

in favor of arithmetic found in the Netherlands during the sixteenth 
century. 16 

In supplementing the proof 17 that the average pressure on a vertical 
square wall of a vessel full of water corresponds to the pressure at its 
mid-point, he gave an "example with demonstration." He subdivided 
the wall into 4 horizontal strips and noted that the force on each is 
greater than 0, tV, ts, and ts units, and less than t*, te, A, and tt 
units respectively; so that the total force is greater than ts and less 
than H- If the wall is subdivided into 10 horizontal strips, the force is 
found similarly to be greater than tou and smaller than rh units; on 
using 1,000 strips, it is determined as more than x^V^oo un i ts > an d 
less than ^""^oVo units. 18 By increasing the number of strips, he then 
remarked, one may approach as closely as desired to the ratio one-half, 
thus proving that the force corresponds to that which would be ob- 
served if the wall were placed horizontally at a depth of half a unit. 19 

This "demonstration by numbers" would correspond exactly to that 
given in the calculus if Stevin had limited himself to one of his two 
sequences and had thought of the results given by his successive sub- 
divisions of the wall as forming literally an infinite sequence with the 
limit \. Stevin, however, shared not only the Greek view with regard 
to infinity but also, although to a lesser extent than did most of his 
mathematical contemporaries, the classical apotheosis of geometry, so 
characteristic of ancient mathematics, which was to prevail in the 
seventeenth century. Even he considered his arithmetic proof, outlined 
above, as merely a mechanical illustration to be distinguished from the 
general mathematical demonstration. 20 

Perhaps the one tendency that did more than any other to conceal 
from mathematicians for almost two centuries the logical basis of the 

16 Cf. Struik, "Mathematics in the Netherlands during the First Half of the XVIth 
Century." 

17 Stevin, Hypomnemata mathetnatica, II, 121 ff. 

a It will be recalled that Stevin had introduced his use of decimal fractions, in De 
Thiende, which had appeared in 1585, the year before the publication of his work in statics. 
See Sarton, "The First Explanation of Decimal Fractions and Measures (1585)." 

19 Stevin, Hypomnemata mathetnatica, II, 125-26; cf. also Bosnians, "Le Calcul chez 
Stevin," pp. 108-9. 

20 "Mathematicae & mechanicae demonstrationis a doctis annotatur differentia, neque 
injuria. Nam ilia omnibus generalis est, & rationem cur ita sit penitus demonstrat, haec 
vero in subjecto duntaxat paradigmate numeris declarat." Stevin, Hypomnemata mathe- 
tnatica, II, 154. 



104 A Century of Anticipation 

calculus was the result of the attempt to make geometrical, rather 
than arithmetic, conceptions fundamental. This will be more true of 
Stevin's successors than it was of him. It must be borne clearly in 
mind, however, that although the logical basis of the calculus is arith- 
metic, the new analysis resulted largely from suggestions drawn from 
geometry. 

The procedures substituted by Stevin for the method of exhaustion 
constituted a marked step toward the limit concept. The extent of his 
influence on contemporary thought, however, is difficult to determine. 
The work in statics which contains his anticipations of the calculus ap- 
peared in Flemish in 1586. It was included also in the Flemish, French, 
and Latin editions of his mathematical works published in 1605-8, and 
in a later French translation of his works by Girard in 1634. 21 With the 
exception of the last-mentioned edition, however, most of these were 
not easily accessible to mathematicians, 22 and, by the time of the French 
translation of 1634, there had already appeared in Italy and Germany 
a number of alternative methods which were destined to become much 
more widely known than those of Stevin. Nevertheless, the influence 
of the Flemish scientist is evident in the thought of a number of later 
mathematicians of the Low Countries. 23 Before we turn to these men, 
however, it may be well to indicate briefly the nature of the modifica- 
tions of Archimedes' work which appeared in Italy and Germany 
shortly after Stevin had published his methods. 

Luca Valerio likewise attempted, in his De centro gravitatis solidorum 
of 1604, a methodization of Archimedes' procedure which should ob- 
viate the need for the reductio ad absurdum and yet retain the neces- 
sary rigor of demonstration. His change was not so sweeping as that of 
Stevin, and his view is not so closely related to the modern. He merely 
tried to substitute for the method of Archimedes a few general theorems 
which could be cited instead of carrying through the details of proof in 
each and every case. No attempt had been made in Greek geometry to 
establish such propositions which might simply be quoted in particular 
cases in lieu of the double reductio ad absurdum demonstration. 24 

a See the article by Sarton, "Simon Stevin of Bruges (1548-1620)"; and that by Bos- 
mans, "Simon Stevin," for biographies of Stevin, analyses of his work, and extensive 
bibliographical references. 

22 Bosnians, "Simon Stevin," p. 889. 

a Bosnians, "Sur quelques exemples de la mSthode des limites chez Simon Stevin." 

u The Works of Archimedes, Introduction, p. cxliii. 



A Century of Anticipation 



105 



Valerio professed that it was the appearance of Commandino's work 
on centers of gravity which had encouraged him, 25 as it had Stevin, 26 
to attempt such a modification of the method of Archimedes. The sig- 
nificant generalization is found in the proposition that, given any figure 
in which the distance between points on opposite sides of a diameter 
vanishes, parallelograms can be inscribed and circumscribed in such a 
way that the excess of the circumscribed figure over that inscribed is 
less than any given area. 27 This is proved by observing that the excess 
is in each case equal to the area of the parallelogram BF (fig. 9) and 
that for suitably chosen approximating figures this "is less than a given 
area." 28 Then Valerio assumed without proof that if the difference be- 




FIGURE 9 

tween the inscribed and the circumscribed figures is smaller than any 
given area, this will be true also of the difference between the curve 
and either of these figures. This geometrical reasoning is strikingly 
similar to that presented in many present-day elementary textbooks 
on the calculus. Valerio did not, however, regard the area of the curve 
as necessarily defined by the limit of the area of either the inscribed 
or the circumscribed figure, as the number of such parallelograms be- 
comes infinite. This is a sophisticated arithmetical conception which 

25 Valerio, De centra gravitalis solidorum libri tres, p. 1. 

26 It is doubtful whether Valerio knew of the work of Stevin. See Bosnians, "Les Demon- 
strations par l'analyse mfinitesimale chez Luc Valerio," p. 211. 

27 Valerio, De centra gravitotis, p. 13. 

28 "Sed parallelogrammum BF est minus superficie proposita." Ibid., p. 14. 



106 A Century of Anticipation 

was not established until two centuries later. Nevertheless, Valerio was 
in a sense anticipating the limit concept, in geometrical form, to the 
extent of indicating the necessary condition for the existence of such a 
limit — viz., that the difference in these areas can be made less than 
any specified area. 

Having generalized, by the above demonstration, the method of in- 
scribing areas, Valerio then proceeded to state general propositions 
which should replace, at least in so far as ratios were concerned, the 
argument by a reductio ad absurdum as found in the method of ex- 
haustion. The intention of these may be stated in the following form: 
If four magnitudes, A, B, C, D, are given, and if two others, G and H, 
can be found which are at the same time greater or smaller than A and 

. G . 

B by a magnitude less than any given magnitude, and if the ratio — is 

H 

at the same time greater or smaller than the ratios — and — , then — 
5 B D B 

C 

= — . 29 This proposition is significant not only as a methodization of 

Archimedes' procedure, but also as a vague striving toward the idea 
which is now expressed concisely by saying that the limit of a ratio of 
two variables is equal to the ratio of the limits of these variables. This 
latter concept is dependent, however, on considerations of the infinite 
— on speculations in which Valerio did not indulge, but which inter- 
ested many of his contemporaries and successors, particularly those 
who combined mathematical investigations with theological interests. 
Johann Kepler had been educated with the intention of entering the 
Lutheran ministry, but had been forced to turn later to the teaching of 
mathematics as a means of earning his livelihood. This may account in 
part for the fact that, although he owed as much to Archimedes as had 
Stevin, the nature of his work is so markedly different from that of the 
engineer of Bruges. There is in the thought of Kepler a deep strain of 
mysticism which was lacking in the attitude of Stevin, of whom Kepler 
may have known. 30 However, this speculative tendency on the part of 

11 De centro gravitatis, p. 69; see also Wallner, "Grenzbegriffes," p. 251. 

30 Bosnians ("Les D6monstrations sur l'analyse infinitesimale chez Luc Valerio," p. 211) 
holds that Kepler was evidently familiar with Stevin's work; and Hoppe ("Zur Geschichte 
der Infinitesimalrechnung," p. 160) that he may have known of it; Wieleitner ("Das 
Fortleben") believes that such an assumption is not justified. 



A Century of Anticipation 107 

Kepler is perhaps with more justice to be ascribed to the Platonic- 
Pythagorean influence, which had been strong in Europe in the pre- 
vious century, 31 and which was so evident in the thought of Cusa, to 
whom Kepler owed at least part of his inspiration. 32 The belief that the 
universe was an ordered mathematical harmony, so strongly shown in 
Kepler's Mysterium cosmographicum, was combined with Platonic and 
Scholastic speculations on the nature of the infinite, giving him a modi- 
fication of Archimedes' mensurational work which was to be a power- 
ful influence in shaping the development of the calculus. 

The ancient Greek philosophers, in their search for unity in this uni- 
verse of perplexing multiplicity, had failed for two reasons to bridge 
the gap between the curvilinear and the rectilinear: first, they banned 
the infinite from geometry; and second, they hesitated, following the 
discovery of the irrational, to pursue further the Pythagorean asso- 
ciation of numerical considerations with geometrical configurations. 
The pious enthusiasm of Kepler, however, saw in this impasse but one 
more evidence of the handiwork of the Creator, who had established all 
things in harmony. God wished quantity to exist so that the comparison 
between a curve and a straight line might be made. This fact was made 
clear to him by the "divine Cusanus" and by others who had regarded 
the forms of the curve and the straight line as complementary, daring 
to compare the curve to God and the line to his creatures. "For this 
reason those who have tried to relate the Creator to his handiwork and 
God to man and divine judgments to human represent an occupation 
which is by no means more useful than that of those who seek to com- 
pare the circle with the square." 33 Under such inspiration, guided as 
well by the speculations of Cusa and Giordano Bruno on the infinite in 
cosmology, 34 and fortified by the knowledge "that nature teaches 
geometry by instinct alone, even without ratiocination," 35 Kepler 
was led to develop his modification of the procedures of Archi- 
medes. 

The task of writing a complete treatise on volumetric determina- 
tions seems to have been suggested to Kepler by the prosaic problem 
of determining the best proportions for a wine cask. The result was 

31 Burtt, Metaphysical Foundations, pp. 44-52; Strong, Procedures and Metaphysics, pp. 
164 ff. 

32 Cf. Kepler, Opera omnia, I, 122; II, 490, 509, 595. 

» Ibid., I, 122. M Ibid., II, 509. « Ibid., IV, 612. 



108 A Century of Anticipation 

the Nova stereometria, which appeared in 1615. 36 This contains three 
parts, of which the first is on Archimedean stereometry, together with 
a supplement containing some ninety-two solids not treated by Archi- 
medes. The second part is on the measurement of Austrian wine barrels, 
and the third on applications of the whole. 

Kepler opened his work on curvilinear mensuration with the simple 
problem of determining the area of the circle. In this he abandoned the 
classical Archimedean procedures. He did not substitute for these the 
limiting consideration proposed by Stevin and Valerio, but had re- 
course instead to the less rigorous but more suggestive approach of 
Nicholas of Cusa. Like Stifel and Viete, he regarded the circle as a 
regular polygon with an infinite number of sides, and its area he there- 
fore looked upon as made up of infinitesimal triangles of which the 
sides of the polygon were the bases and the center of the circle the 
vertex. The totality of these was then given by half the product of 
the perimeter and the apothem (or radius). 37 

Kepler did not limit himself to the simple proposition above, but 
with skill and imagination applied this same method to a wide variety 
of problems. By looking upon the sphere as composed of an infinite 
number of infinitesimal cones whose vertices were the center of the 
sphere and whose bases made up the surface, he showed that the vol- 
ume is one-third the product of the radius and the surface area. 38 The 
cone and cylinder he regarded variously: as made up of an infinite 
number of infinitely thin circular laminae (as had Democritus two 
thousand years earlier), as composed of infinitesimal wedge-shaped 
segments radiating from the axis, or as the sum of other types of 
vertical or oblique sections. 39 The volumes of these he computed by 
the application of such views. In a similar manner he rotated a circle 
about a line and calculated, by infinitesimal methods, the volume of 
the anchor ring, or tore, thus generated. 40 This determination was 
equivalent to an application, for a special case, of the classical theorem 
of Pappus, later called Guldin's rule. Kepler then extended his work 
to solids not treated by the ancients. The areas of the segments cut 
from a circle by a chord he rotated about this chord, obtaining solids 

36 Followed, about a year later, by a popular German edition. The Latin appears, with 
notes, in Volume IV, Opera omnia; the German in Volume V. 

37 Opera omnia, IV, 557-58. 

» Ibid., IV, 563. 3 » Ibid., IV, 564, 568, 576 ff. « Ibid., IV, 575-76, 582-83, 



A Century of Anticipation 109 

which he designated characteristically as apple or citron-shaped, ac- 
cording as the generating segment was greater or less than a semi- 
circle. 41 The volumes of these and other solids he likewise calculated 
by his infinitesimal methods. To Willebrord Snell, the editor of Stevin's 
works, Kepler proposed, as a challenge, the determination of the solids 
obtained similarly by the rotation of segments of conic sections. 42 This 
problem was significant in the later work of the Italian mathematician 
Cavalieri. 

Some of Kepler's summations are remarkable anticipations of re- 
sults found later in the integral calculus. 43 In his well-known Astro- 
nomia nova of 1609, for example, there is a computation 44 resembling 
that which is expressed in modern notation as Jo sin 6 dd = 1 — cos 6. 
Other calculations in this work correspond to approximations to elliptic 
integrals, 45 in one of which v(a + b) is given as the approximate length 
of the ellipse with the semi-axes a and b. 46 Kepler, however, was far 
from clear on the point of the basic conceptions involved, with the re- 
sult that his work is not free from errors. 47 He generally spoke of sur- 
faces and volumes as made up of infinitesimal elements of the same 
dimension, but occasionally he lapsed into the language of indi- 
visibles, which his successor Cavalieri was to find so congenial. In one 
place he spoke of the cone as though composed of circles, 48 and in the 
work by which he arrived in his astronomy at his famous second law 
he regarded the sector of an ellipse as the sum of its radius vectors. 49 

Kepler appears not to have distinguished clearly between proofs by 
means of the method of exhaustion, by ideas of limits, by infinitesimal 
elements, or by indivisibles. The conceptions which he held in his dem- 

a Wolf (History of Science, Technology, and Philosophy in the Sixteenth and Seventeenth 
Centuries, pp. 204-5) has correctly pointed out that the word citrium, which Kepler used 
in this connection, is properly translated as "gourd"; but he has failed to add that Kepler 
himself, in his German edition, translated it as "citron." Opera omnia, V, 526. 

"Ibid., IV, 601,656. 

43 See Struik, "Kepler as a Mathematician," in Johann Kepler, 1571-1630. This volume 
also contains an excellent bibliography of Kepler's works, by F. E. Brasch. 

44 See Opera omnia, III, 390; cf. also Gunther, "Uber ein merkwiirdige Beziehung 
zwischen Pappus und Kepler"; Enestrom, "Uber die angebliche Integration einer trig- 
onometrischen Funktion bei Kepler." 

45 Cf. Struik, "Kepler as a Mathematician," p. 48; Zeuthen, Geschichte der Mathematik 
im XVI. und XVII. Jahrhunderl, pp. 254-55. 

46 Opera omnia, III, 401. 47 Moritz Cantor, Vorlesungen, II, 753. 

48 "Nam conus est hie veluti circulus corporatus." Opera omnia, IV, 568. 

49 Opera omnia, III, 402-3. 



110 A Century of Anticipation 

onstrations are a far cry from the notions held by the ancient geom- 
eters. The Greek thinkers saw no way of bridging the gap between the 
rectilinear and the curvilinear which would at the same time satisfy 
their strict demands of mathematical rigor and appeal to the clear 
evidence of sensory experience. Fortified by the scholastic disputations 
on the categorical infinite and by Platonic mathematical speculations, 
Kepler followed Nicholas of Cusa in resorting to a vague "bridge of 
continuity" which finds no essential difference between a polygon and 
a circle, between an ellipse and a parabola, between the finite and the 
infinite, between an infinitesimal area and a line. 50 This striving for an 
expression of the idea of continuity constantly reappears throughout 
the period of some fifty years preceding the formulation of the methods 
of the calculus. Leibniz himself, like Kepler, frequently fell back upon 
his so-called law of continuity when called upon to justify the differ- 
ential calculus; and Newton concealed his use of the notion of con- 
tinuity under a concept which was empirically more satisfying, though 
equally undefined — that of instantaneous velocity, or fluxion. 

Kepler's Doliometria, or Stereometria doliorum, exerted such a strong 
influence in the infinitesimal considerations which followed its ap- 
pearance, and which culminated a half century later in the work of 
Newton, that it has been called, with perhaps pardonable exaggeration, 
the source of the inspiration for all later cubatures. 81 Before passing on 
to these anticipations of the integral calculus, there should be cited a 
contribution which Kepler made to the thought leading to the differ- 
ential calculus. The subject of the measurement of wine casks had led 
Kepler to the problem of determining the best proportions for these. 62 
This brought him to the consideration of a number of problems on 
maxima and minima. In the Doliometria he showed, among other 
things, that of all right parallelepipeds inscribed in a sphere and having 
square bases, the cube is the largest; 53 and that of all right circular 
cylinders having the same diagonal, that one is greatest which has the 
diameter and altitude in the ratio of V2 to l. 54 

These results were obtained by making up tables in which were 

60 Cf . Taylor, "The Geometry of Kepler and Newton." 

61 Moritz Cantor, Vorlesungen, II, 750. 

62 Kepler discovered, incidentally, that the Austrian barrels approximated very closely 
the desired proportions. 

» Opera omnia, IV, 607-9. M Ibid; IV, 610-12. 



A Century of Anticipation 111 

listed the volumes for given sets of values of the dimensions, and from 
these selecting the best proportions. An inspection of such tables 
showed him an interesting fact. He remarked that as the maximum 
volume was approached, the change in the volume for a given change 
in the dimensions became smaller. Oresme, several centuries earlier, 
had made a similar observation, but had expressed it differently. 
Oresme had noticed that for a form which was represented graphically 
by a semicircle, the rate of change was least at the maximum point. 
This thought appeared again, in the seventeenth century, in the 
methods of the French mathematician Fermat. Whether Fermat was 
influenced in this direction by Kepler or Oresme is problematical; but 
a comparison of the distinctly different points of view which the latter 
men represent will aid later in understanding Fermat's approach. 
Kepler had made his remark upon the basis of numerical considera- 
tions. He was, moreover, more particularly concerned with static con- 
siderations as found in Greek geometry and in methods of indivisibles. 
He consequently expressed himself in terms of increments and decre- 
ments near the maximum point. On the other hand, the medieval 
problem of the latitude of forms and the graphical representation of 
continuous variability had led Oresme to state the conclusion in terms 
of the rate of change. The latter view has been made fundamental in 
mathematics through the concept of the derivative; but it is Kepler's 
mode of expression which appeared in the work of Fermat. Although 
the Scholastic views on variation played a significant role in the an- 
ticipations of the calculus, the static approach of Kepler predominated. 
Increments and decrements, rather than rates of change, were the 
fundamental elements in the work leading to that of Leibniz, and 
played a larger part in the calculus of Newton than is usually recog- 
nized. The differential became the primary notion and it was not ef- 
fectively displaced as such until Cauchy, in the nineteenth century, 
made the derivative the basic concept. 

Twenty years after the publication of the Stereometria doliorum of 
Kepler there appeared in Italy a work which rivaled it in popularity. 
So famous did the Geometria indivisibilibus of Bonaventura Cavalieri 
become that it has been maintained, with some justice, that the new 
analysis took its rise from the appearance, in 1635, of this book. 55 To 

68 Leibniz, The Early Mathematical Manuscripts, trans, by Child, p. 196. 



112 A Century of Anticipation 

what extent this is indebted to the earlier work of Kepler is difficult to 
determine. Cavalieri emphatically denied any inspiration from Kepler's 
method, other than "the names of a few solids and the admiration 
which frequently sets philosophers to reflecting." 56 It is not improbable, 
however, that the influence of Kepler upon Cavalieri may have resulted 
indirectly from the correspondence of both of these men with Galileo. 67 
A work upon indivisibles which Galileo planned to write never ap- 
peared, but his views upon the subject are clearly brought out in his 
classic treatise, the Two New Sciences, which was published three years 
after the G'eometria of Cavalieri. Galileo's opinions resemble strongly 
those expressed by his pupil, Cavalieri, and may well have been the 
source of the latter's inspiration. 68 It may be well, therefore, to examine 
at this point the views of Galileo, the teacher. 

The forces molding the thought of Galileo and Cavalieri did not 
differ greatly from those which had shaped the ideas of Kepler. These 
men had mastered the Greek geometric methods, but they all betray 
the effects of Scholastic speculations and the Platonic view of mathe- 
matics which had exerted such a strong influence since the time of 
Nicholas of Cusa. Both Galileo and Cavalieri were probably acquainted 
with the modifications of the method of Archimedes by Valeric 
Galileo referred to Valerio several times, in the Two New Sciences, as 
the great geometer and as the new Archimedes of his age. 69 Galileo 
gave in this work also an Archimedean demonstration of the quadra- 
ture of the parabola 60 and included in an appendix some work on 
centers of gravity in the manner of Commandino and Valerio. 61 
Nevertheless both he and Cavalieri appear to be more significantly in- 
debted, in attitude and method, to the later medieval speculations on 
motion, indivisibles, the infinite, and the continuum. 62 

The influence of Scholastic thought is clearly evidenced in the case 

of Galileo by his early writings. In these he considered, among other 

things, the Peripatetic doctrines of matter and form and of causes and 

qualities, and the Scholastic questions of intension and remission and 

of action and reaction. 63 On the latter he referred specifically to works 

68 Exercitationes geometricae sex, pp. 237-38. 

"Paul Tannery, Notions hisforiques, p. 341; Moritz Cantor, Vorlesungen, II, 774-75. 
68 Paul Tannery, ibid., p. 341. 69 Le open di Galileo Galilei, VIII, 76, 184, 313. 

•o Opere, Will, 181 ff . " Opere, VIII, 313 ; I, 187-208. 

M Wallner, "Die Wandlungen des Indivisibiliensbegriffs von Cavalieri bis Wallis." 
"Opere, I, "Iuvenalia," 111 ff., 119 ff., 126 ff. 



A Century of Anticipation 



113 



by Calculator and Hentisbery. 64 In dynamics he made fundamental 
the doctrine of impetus, which had been suggested by the Scholastic 
philosopher Buridan in the fourteenth century and which was cer- 
tainly familiar to Nicholas of Cusa, Leonardo da Vinci, and others of 
the fifteenth and sixteenth centuries. 65 Whether Cusa's influence on 
Galileo was significant (as it was on Kepler and probably also on 
Giordano Bruno) is difficult to determine; 66 but that the Scholastic 
discussions on motion were known to him will be very strongly indi- 
cated by an examination of one of the propositions in his Two New 
Sciences. 



E 



G 












j 








/ 




/ 




/ 





F 
FIGURE 10 



D 



It will be recalled that Calculator and Hentisbery had demonstrated 
dialectically that the average velocity of a body moving with uniform 
acceleration is given by its velocity at the mid-point of the time in- 
terval. Oresme had given a geometrical demonstration of this propo- 
sition, in which he indicated that the area under the line representing 
the velocity was the measure of the distance. Galileo's demonstration 
of this proposition resembles strikingly that of Oresme. Let AB (fig. 10) 
represent the time in which the space CD is traversed by a body which 

84 IMA-, I, 172. •« Duhem, Etudes sur Leonard de Vinci, Vol. Ill, passim. 

•• Goldbeck, "Galileis Atomistik und ihre Quellen." 



114 A Century of Anticipation 

starts from rest and is uniformly accelerated. Let the final speed be 
represented by EB. Then the lines drawn parallel to EB will represent 
the speeds of the body. It appears, then, that they may be interpreted 
also as the moments, or infinitesimal increments, in the distance 
covered by the moving body. Then the movements of the uniformly 
accelerated motion may be represented by the parallels of triangle 
AEB, whereas the parallels of the rectangle ABFG represent the cor- 
responding moments of a body moving uniformly. But the sum of all 
the parallels contained in the quadrilateral ABFG is equal to the sum 
of those contained in the triangle A EB. Hence it is clear that the dis- 
tances covered by the two bodies are equal, inasmuch as the triangle 
and the rectangle are equal in area if / is the midpoint of FG. 67 

Not only did Galileo reproduce with striking fidelity the diagram 
and argument of Oresme as outlined above, but he extended a remark 
made in this connection by Calculator and Hentisbery. Galileo could 
have read in the works of these men, or in commentaries upon them, 
that in uniformly accelerated motion the space covered in the second 
half of the time is three times that covered in the first half. 68 This ob- 
servation Galileo extended to show that if we subdivide the time in- 
terval further, the distances covered, in each of these, will be in the 
ratio 1, 3, 5, 7, ... 69 This is, of course, equivalent to the result ex- 
pressed by the formula 5 = %gt 2 , and is implied by the earlier work of 
the Scholastics. 

The difference between Galileo's demonstration and that of Oresme 
is largely one of completeness. Oresme had been satisfied to say merely 
that inasmuch as the triangles EIF and AIG are equal, the distances 
must be the same, thus implying the infinitesimal considerations neces- 
sary to demonstrate that the areas represent distances. The geometrical 
demonstrations of Oresme and Galileo are based upon the supposition 
that the area under a velocity-time curve represents the distance 
covered. 70 Since neither one possessed the limit concept, each resorted, 
explicitly or implicitly, to infinitesimal considerations. Galileo ex- 
pressed this view when he said that the moments, or small increments 
in the distance, were represented by the lines of the triangle and the 

*Opere, VIII, 208-9. 

«» Duhem, Etudes sur Leonard de Vinci, III, 480, 513. 69 Opere, VIII, 210 ff. 

70 Paul Tannery, "Notions historiques," pp. 338-39, mistakenly attributes this idea to 
Galileo, rather than Oresme. 



A Century of Anticipation 115 

rectangle, and that these latter geometrical figures were in actuality- 
made up of these lines. Galileo did not make clear how the transition 
from the lines as velocities to the same lines as moments is to be made. 
Oresme likewise had begged the question when he had represented in- 
stantaneous velocities by lines and yet had maintained that all ve- 
locities act through a time. Galileo and Oresme patently employed the 
uncritical mathematical atomism which has appeared among mathe- 
maticians of all ages — in Democritus, Plato, Nicholas of Cusa, Kepler, 
and many others. It has been suggested that Galileo was influenced in 
his use of indivisibles by his strong Pythagorean and Platonic approach 
to science, 71 or by the revival in his day of interest in the atomism of 
Heron of Alexandria, in which Galileo failed to distinguish clearly be- 
tween physical indivisibles and mathematical infinitesimals. 72 He may 
equally well have been led to his views by the Scholastic discussions of 
infinitesimals. 

At any rate, in the dialogue of the first day, in the Two New Sciences, 
Galileo entered into an extended discussion of the subjects which had 
been so popular in the medieval period: the infinite, the infinitesimal, 
and the nature of the continuum. Galileo clearly admitted the possi- 
bility of the Scholastic categorematic infinity, but because of the nu- 
merous paradoxes to which this appeared to lead, Galileo concluded 
that "infinity and indivisibility are in their very nature incompre- 
hensible to us." 73 Nevertheless, he made at least one trenchant ob- 
servation upon this subject. It will be recalled that Calculator had 
remarked that there can be no ratio between an infinite magnitude and 
a finite one. Galileo asserted more generally "that the attributes 
'larger,' 'smaller,' and 'equal' have no place either in comparing in- 
finite quantities with each other or in comparing infinite with finite 
quantities." 74 In justifying this conclusion, Galileo indicated a sig- 
nificant shift of emphasis, for instead of considering the infinite from 
the point of view of magnitude, as had Aristotle and many medieval 
scholars, he focused attention, as had Plato, upon the infinite as 
multiplicity or aggregation. In this connection he indicated that the 

"Wiener, "The Tradition behind Galileo's Methodology"; cf. also Brunschvicg, Les 
lilapes de la philosophie mathematique, p. 70. 

72 See Schmidt, "Heron von Alexandria im 17. Jahrhundert"; also other articles on 
Heron in the same volume. 

73 Opere, VIII, 76 ff . » Opere, VTH, 82 ff . 



116 A Century of Anticipation 

infinite class of all positive integers could be put into a one-to-one cor- 
respondence with a part of this class — for example, with all the perfect 
squares. 76 This characteristic of infinite sets was rediscovered in the 
nineteenth century by Bolzano and later in that century was made 
fundamental in the establishment of the calculus upon a rigorously 
developed theory of infinite assemblages. 76 

In spite of the trenchant observations which Galileo made on the 
subject of the infinite, he felt strongly the inability of intuition to grasp 
this notion. He went so far as to conjecture that there might be a third 
possible type of aggregation between the finite and the infinite. This 
seems to have been suggested to him by the difficulties of making pre- 
cise our vague ideas on the continuum. He maintained, contrary to 
Bradwardine, that continuous magnitudes are made up of indivisibles. 
However, inasmuch as the number of parts is infinite, the aggregation 
of these is not one resembling a very fine powder but rather a sort of 
merging of parts into unity, as in the case of fluids. 77 This analogy is a 
beautiful illustration of the effort which men made to picture in some 
way the transition from the finite to the infinite. 

Galileo sought also to clarify to some extent the paradoxes on mo- 
tion. This he did by regarding rest as an infinite slowness, thus resort- 
ing again to the vague feeling for the continuous to which Cusa and 
Kepler had sought to give expression. In enlarging upon this idea, 
Galileo applied to a falling body the argument which Zeno had given 
in the dichotomy and then answered this by an appeal to intuition in 
reversing the descent. In ascent, the body passes through an infinite 
number of grades of slowness, finally to come to rest. The beginning 
of motion is precisely the same, except that the order is reversed. 78 
This argument constitutes a recognition — found also in Aristotle — of 
the similarity of the difficulties in the dichotomy and the Achilles. It 
represents, as well, an attempt to clarify the sense in which an infinite 
series may be said to have a sum. Later Newton in his calculus, when 
he spoke of his instantaneous velocities as prime or ultimate ratios, 
made an analogous appeal to the fact that the beginning and the end 
of motion are to be similarly conceived. Galileo appears not to have 
realized, as Newton did somewhat vaguely, that only in terms of the 

™Ibid.,Vni, 78. 

76 Kasner, "Galileo and the Modern Concept of Infinity," pp. 499-501. 

77 Opere, VTII, 76 ff . n Opere, VIII, 199-200. 



A Century of Anticipation 117 

limit concept can precise meaning be given to either the sum of an 
infinite series or to a first or last ratio. 

Whatever the influences which shaped the thought of Galileo, those 
felt by his friend and student, Cavalieri, are not likely to have been 
greatly different. Although familiar with the views of Valerio and 
perhaps also with those of Stevin, 79 Cavalieri did not develop the limit 
idea which these men had adumbrated. Instead he had resort to the 
less subtle notion of the indivisible which had been adopted by Galileo. 80 
However, whereas Galileo had employed this in physical explanation, 
Cavalieri made it the basis of a geometrical method of demonstration 
which achieved remarkable popularity. This method had been de- 
veloped by Cavalieri as early as 1626, for in that year he wrote to 
Galileo, saying that he was going to publish a book on the subject. 81 
This work appeared in 1635, as the Geometria indivisibilibus continu- 
orum nova quadam ratione promote,* 2 and the method it presented 
was further developed a dozen years later in the Exercitationes geome- 
tricae sex. 

Cavalieri at no point in his books explained precisely what he under- 
stood by the word indivisible, which he employed to characterize the 
infinitesimal elements used in his method. He spoke of these in much 
the same manner as had Galileo in referring to the parallel lines rep- 
resenting velocities or moments as making up the triangle and the 
quadrilateral. Cavalieri conceived of a surface as made up of an indefi- 
nite number of equidistant parallel lines and of a solid as composed of 
parallel equidistant planes, 83 these elements being designated the in- 
divisibles of the surface and of the volume respectively. Although he 
recognized that the number of these must be indefinitely great, he did 
not follow his master Galileo in speculations as to the nature of the 
infinite. The attitude of Cavalieri toward infinity was one of agnosti- 
cism. 84 He did not share the Aristotelian view of infinity as indicating 
a potentiality only — a conception which, in conjunction with the work 
of Stevin and Valerio, pointed toward the method of limits. On the 
other hand, Cavalieri did not join Nicholas of Cusa and Kepler in re- 

79 Bosmans, "Les Demonstrations par l'analyse infinitesimale," p. 211. 

80 Marie, Histoire des sciences mathematiques et physiques, III, 134. 
a Moritz Cantor, Vorlesungen, II, 759. 

82 1 have used the later edition, Bononiae, 1653. 

85 Exercitationes geometricae sex, p. 3. 

84 Cf. Brunschvicg, Les tiapes de la philosophic maihematique, p. 166. 



118 A Century of Anticipation 

garding the infinite as possessing a metaphysical significance. It was 
employed by him solely as an auxiliary notion, comparable to the "so- 
phistic" quantities of Cardan. Inasmuch as it did not appear in the 
conclusion, its nature need not be made clear. That the infinite did 
not enter explicitly into the arguments of Cavalieri was due to the fact 
that at every stage he centered attention upon the correspondence 
between the indivisibles of two configurations, rather than upon the 
totality of indivisibles within a single area or volume. The proposition 
still known in solid-geometry textbooks as Cavalieri' s Theorem is char- 
acteristic of his approach: If two solids have equal altitudes, and if 
sections made by planes parallel to the bases and at equal distances 
from them are always in a given ratio, then the volumes of the solids 
are also in this ratio. 86 



FIGURE 11 

Typical of the propositions in the method of indivisibles, and of far- 
reaching significance in later developments, were a number of theorems 
on the lines of a parallelogram and those of its constituent triangles. 
One of these propositions consisted in an extended demonstration that 
if a parallelogram AD (fig. 11) is divided by the diagonal CF into two 
triangles, ACF and DCF, then the parallelogram is double either 
triangle. 86 This is proved by showing that if one lay off EF = CB and 
draw HE and BM parallel to CD, then the lines HE and BM are equal. 
Therefore all the lines of triangle ACF taken together are equal to all 
those of CFD; consequently triangles ACF and CDF are equal, and 

w Geometria indivisibilibus, pp. 113-15; Exercitationes geometricae sex, pp. 4-5. Cf. also 
Evans, "Cavalieri's Theorem in His Own Words." 

M Geotnetria indivisibilibus, Proposition XIX, pp. 146-47; Exercitationes geometricae sex, 
Proposition XIX, pp. 35-36. 



A Century of Anticipation 119 

the sum of the lines of the parallelogram AD is double the sum of the 
lines of either triangle. 

From here Cavalieri went on to prove by a similar, but considerably 
more involved, argument that the sum of the squares of the lines in the 
parallelogram is three times the sum of the squares of the lines in each 
of the constituent triangles. 87 Using this latter proposition, he then 
easily demonstrated, among other things, that the volume of a cone is 
§ that of the circumscribed cylinder, and that the area of a parabolic 
segment is § the area of the circumscribed rectangle. 88 These results 
were of course known to Archimedes, but a problem which Kepler had 
proposed some years before now led Cavalieri to a use of indivisibles 
bolder than that which is found in the Geometric, and to a new result 
of significance. 89 

Kepler had, in his Stereometria, challenged geometers to find the 
volume of the solid obtained by rotating a segment of a parabola about 
its chord. 90 Cavalieri determined this volume by basing the problem 
on the discovery that the sum of the fourth powers of the lines of a 
parallelogram is 5 times the sum of the fourth powers of the lines of 
one of the constituent triangles. Then he recalled that in his Geometria 
indivisibilibus he had found the ratios of the lines to be 2 to 1 and the 
ratio of the squares of the lines to be 3 to 1. In order not to leave a gap 
in his results on the ratios of the powers of the lines of a parallelogram 
and of the triangle, Cavalieri sought the ratio for the sums of the cubes 
of the lines, and found this to be as 4 to 1. He then concluded by anal- 
ogy that for the fifth powers it would be 6 to 1, for sixth powers 7 to 
1, and so on, the sum of the nth powers of the lines of the parallelogram 
being to the sum of the nth powers of the lines of the triangle as n + 
1 to l. M 

The method of Cavalieri here employed was based upon several 
lemmas which are equivalent to special cases of the binomial theorem. 92 
For example, to prove that the sum of the cubes of the lines of a paral- 

87 Geometria indivisibilibus, Proposition XXIV, pp. 159-60; Exercitationes geometricae 
sex, Proposition XXIV, pp. 50-51. 

88 Geometria indivisibilibus, pp. 185, 285-86; Exercitationes geometricae sex, pp. 78 ff. 

89 For a summary of this work, see Bosmans, "Un Chapitre de l'oeuvre de Cavalieri." 

80 Opera omnia, IV, 601. 

81 Exercitationes geometricae sex, pp. 290-91. For a statement of the manner in which he 
was led to this result, see pp. 243-44. 

92 Exercitationes geometricae sex, p. 267. 



120 A Century of Anticipation 

lelogram is four times the sum of the cubes of the lines of one of the 
constituent triangles, he began with (a -f b) 3 = a 3 + 3a 2 b + 3ab 2 -f b 3 . 
Then he proceeded somewhat as follows: setting AF = c, GH = a, 
HE = b, in figure 11, we have 2c 3 = 2a 3 + 32a 2 6 + 32ab 2 + 2& 3 , 
where the sums are taken over the lines of the parallelogram and tri- 
angles. By symmetry this can be written 2c 3 = 22a 3 + 62a 2 6. Now 
2c 3 = cZc 2 - c2 (a + b) 2 = cZa? + ICLab + c2& 2 . But in an earlier 
proposition on lines Cavalieri had shown that 2a 2 = 2& 2 = \l>c 2 . This 
gives us 2c 3 = |c2c 2 ■+ 2c2a& 

= §2c 3 + 2 (a + b)2ab = |2c 3 + 22a 2 6 + 22& 2 a 

= 12c 3 + 42a 2 6 

or 2a 2 6 = tV2c 3 . Substituting this in the equation above, we obtain 

2c 3 = 22a 3 + ^2c 3 , or 2c 3 = 42a 3 , and the proposition is proved. 93 

Cavalieri realized that this method can be generalized for all values 

of «, 94 but he gave complete demonstrations only up to and including 

n — 4. For higher values he gave only some "cossic" indications, which 

had been given him by Beaugrand. 95 These may well have been derived 

from the contemporaneous work of Fermat, 96 whose methods we shall 

consider later. 

The results of Cavalieri outlined above are in a general sense equiva- 

a n + 1 

lent to what we would now express by the notation J x dx = . 

n + 1 

We have seen that Archimedes, through the use of series in connection 
with his stereometric work, had recognized the truth of this proposi- 
tion for n = 1 and n = 2. He may have known of it for n = 3 also, and 
the Arabs proved it for n = 4 as well. 97 Cavalieri's work, although 
based upon somewhat different views, was a generalization of these 
results of Archimedes and the Arabs. It will be seen later that in this 
respect Cavalieri had been anticipated by several contemporary 
mathematicians. Nevertheless, his statement, which appeared in 1639, 98 
represents the first publication of this theorem, which played a signifi- 

» Ibid., pp. 273-74. M "Et sic in infinitum." Ibid., p. 268. 

95 Ibid., pp. 286-89. 9 « See Fermat, CEuvres, Supplement, p. 144. 

87 See Paul Tannery, "Sur le sommation des cubes entiers dans l'antiquitfi," and Ibn 
al-Haitham (Suter), "Die Abhandlung uber die Ausmessung des Paraboloids." 

88 Cenluria di varii problemi, pp. 523-26. Simon ("Zur Geschichte und Philosophic der 
Differentialrechnung," p. 118) has said — very probably through some error — that Cava- 
lieri found this result in 1615. 



A Century of Anticipation 121 

cant role in the development of infinitesimal methods during the period 
from 1636 to 1655. Within this interval, the mathematicians Torricelli, 
Roberval, Pascal, Fermat, and Wallis arrived, all more or less inde- 
pendently and by varying methods, at this fundamental result and 
extended it, as well, to include negative, rational fractional, and even 
irrational values of n. It was perhaps the first theorem in infinitesimal 
analysis to point toward the possibility of a more general algebraic 
rule of procedure, such as that which, formulated a generation later 
by Newton and Leibniz, became basic in the integral calculus. Cava- 
lieri himself had no vision of such a new analysis; neither he nor Galileo 
appears to have been seriously interested in algebra, either as a manner 
of expression or as a form of demonstration. The proposition remained 
for Cavalieri a geometrical theorem concerning the ratio subsisting be- 
tween a sum of powers of the lines of a parallelogram and that of one 
of the constituent triangles. Furthermore, he did not express any clear 
conception — such as is basic in our idea of the definite integral — of 
his ratios as limits of sums of infinitesimal parallelograms. Cavalieri 
never did make clear his interpretation of the indivisible and in this 
respect laid his method open to attack. 

Cavalieri's use of indivisibles in his Geometria had been criticized by 
the Jesuit, Paul Guldin, who asserted not only that the method had 
been taken from Kepler, but also that it was incorrect, inasmuch as it 
led to paradoxes and fallacies. Cavalieri defended himself from the 
first charge by pointing out that his method differed from that of 
Kepler in that it made use only of indivisibles, whereas Kepler had 
thought of a solid as made up of very tiny solids." In answering the 
charge that the method was invalid, Cavalieri maintained that al- 
though the indivisibles may correctly be considered as having no 
thickness, nevertheless, if one wished, he could substitute for them 
small elements of area and volume in the manner of Archimedes. 
Guldin had said that since the number of indivisibles was infinite, 
these could not be compared with one another. Furthermore, he had 

w "Ex minutissimis corporibus." Exercitationes geometricae sex, p. 181; see also Kepler, 
Opera omnia, IV, 656-57, for notes, including extracts from Cavalieri on this point. 
Cavalieri's assertion in this connection should be sufficient to refute the statement often 
made (see e. g., Wolf, A History of Science, Technology, and Philosophy in the Sixteenth and 
Seventeenth Centuries, p, 206) that he was probably well aware of the fact that his indi- 
visibles must be of the same dimension as that of the figure which they constitute. 



122 A Century of Anticipation 

pointed out a number of fallacies to which the method of indivisibles 
appeared to lead. In answering these arguments, Cavalieri said that 
difficulties in the method are avoided by observing that the two 
infinities of elements to be compared are of the same kind. If, for 
example, the altitudes of two figures are unequal, their horizontal sec- 
tions are not to be compared, because the corresponding indivisibles 
of one are not the same distance apart as are those in the other. Where- 
as in one figure there may be 100 indivisibles between two sections, 
between the corresponding sections of the other there may be 200. ,0 ° 

This explanation Cavalieri followed with an ingenuous comparison 
of the indivisibles of a surface with the threads of a piece of cloth, and 
those of a solid with the pages of a book. 101 Although in geometrical 
solids and surfaces the indivisibles are infinite in number and lacking 
in all thickness, nevertheless they may be compared in the same manner 
as in case of the cloth and the book, if one observes the precaution 
mentioned above. Cavalieri did not explain how an aggregate of ele- 
ments without thickness could make up an area or volume, although 
in a number of places he linked his method of indivisibles with ideas 
of motion. This association had been suggested somewhat vaguely by 
Plato and the Scholastic philosophers, and Galileo had followed them 
in associating dynamics with geometrical representation. Napier, in 
1614, had likewise employed the idea of the fluxion of a quantity to 
picture by means of lines the relation between logarithms and num- 
bers. 102 Cavalieri followed this trend in holding that surfaces and 
volumes could be regarded as generated by the flowing of indivisibles. 103 
He did not, however, develop this suggestive idea into a geometrical 
method. This was done by his successor Torricelli, with the result that 
it eventuated later in Newton's method of fluxions. Cavalieri's indi- 
visible itself was also to find a counterpart in the thought of both 
Newton and Leibniz — in the former's conception of moments and in 
the latter's notion of differentials. Cavalieri's vague suggestions were 
thus to play a large part in the development of the calculus. 

The Geometria indivisibilibus of Cavalieri achieved popularity al- 
most immediately, and became, except for the works of Archimedes, 

100 Exercitationes geometricae sex, pp. '238-39; cf. also p. 17. 

101 Ibid., pp. 239-40; cf. also pp. 3-4. 

m Zeuthen, Geschichte der Mathematik im XVI. und XVII. Jahrhundert, pp. 134-35. 
108 Exercitationes geometricae sex, pp. 6-7; Geometria indivisibilibus, p. 104. 



A Century of Anticipation 123 

the most quoted source for mathematicians dealing with infinitesimal 
considerations in geometry. The significance of the work in the history 
of mathematics has been justly recognized, but such recognition has 
on occasion led to attempts to impute to Cavalieri views which he was 
far from possessing. It has been asserted that in his work one discerns 
clearly "the fundamental notion of the differential calculus," inasmuch 
as "the indivisible is nothing but the differential," 104 and that one dis- 
covers in it "definite integrals in the sense of Cauchy and Riemann." 108 
An examination of his ideas and methods will show that such judg- 
ments are not sound. Cavalieri was far from possessing the views which 
are expressed in the terms "differential" and "integral." He himself 
appears to have regarded his method only as a pragmatic geometrical 
device for avoiding the method of exhaustion; the logical basis of this 
procedure did not interest him. Rigor, he said, was the affair of phi- 
losophy rather than geometry. 106 The limit idea, toward which Stevin 
and Valerio were working, was more completely concealed in Cava- 
lieri's method than in Kepler's. Furthermore, there is in the Geometria 
indivisibilibus a complete lack of emphasis on the algebraic and 
arithmetical elements which were to lead, first to the rules of procedure 
of the calculus and later to the satisfactory definitions of the differ- 
ential and the integral. Cavalieri regarded area and volume as in- 
tuitively clear geometrical concepts, and invariably determined the 
ratio of these, rather than a numerical value associated with a single 
area or volume. This preoccupation with ratios was to be one of the 
chief causes of the confusion in the basic ideas of the calculus during 
the following two centuries. 

The fact that Cavalieri paid so little heed to the demands of mathe- 
matical rigor made geometers chary of accepting the method of in- 
divisibles as valid in demonstrations, although they employed it 
readily in preliminary investigations. This element of hesitation is seen 
to particularly good advantage in the work of Evangelista Torricelli, 
the friend of Cavalieri and the pupil of Galileo. 

Torricelli fully realized the advantages and disadvantages of the 
method of indivisibles; and he suspected that the ancients possessed 

104 Milhaud, "Note sur les origines du calcul infinitesimal," pp. 37-38. 
106 Bortolotti, "La scoperta e le successive generalizzazioni di un teorema fondamentale 
di calcolo integrale," p. 210, n. 

108 Exercitationes geometricae sex, p. 241. 



124 A Century of Anticipation 

some such method for discovering difficult theorems, the proofs of 
which they cast in another form either "to hide the secret of their 
method or to avoid giving occasion for contradiction to jealous de- 
tractors." 107 The work of Archimedes indicates how correct Torricelli 
was in his assumption of the existence of this method, but the motive 
behind the proofs by exhaustion lay rather in an effort to satisfy the 
Greek demands for intuitive clarity and logical rigor. Torricelli himself 
was not quite satisfied with demonstrations by the method of indi- 
visibles, for he usually supplemented these by proofs in the manner of 
Archimedes, or of Valerio, to whom he refers (echoing Galileo) as the 
"Archimedes of our century." 108 

The De dimensions parabolae of Torricelli is an interesting exercise, 
in which the author offers twenty-one demonstrations of the quadra- 
ture of the parabola. In ten of these the proposition is established by 
the method of the ancients, 109 and in the other eleven by the new 
geometry of indivisibles. 110 Included among the former 111 is the well- 
known demonstration by the method of exhaustion, given by Archi- 
medes in his Quadrature of the Parabola. There is, as well, a proof 112 
closely resembling the basic proposition of Luca Valerio on inscribed 
and circumscribed figures; viz., that it is possible to inscribe within 
the parabolic segment a figure, made up of parallelograms of equal 
height, which shall differ from the segment by less than any given 
magnitude*. Torricelli did not, of course, consider the area of the parab- 
ola as defined by the limit of the inscribed figure, but he approached 
more closely to this idea than had Valerio. The latter had been satisfied 
to state that the circumscribed and inscribed figures differed by less 
than a given magnitude and to imply that the figure itself would there- 
fore differ from either of these by less than this magnitude; Torricelli 
clearly stated this implication, from which the limit concept would 
easily follow on arithmetizing the quantities involved. However, Torri- 
celli followed Cavalieri in restricting himself to geometrical considera- 
tions, and in consequence he was attracted more toward indivisibles 
than toward limits. 

107 Opere di Evangdista Torricelli, I (Part 1), 140. »* Opere, I (Part 1), 95. 

io» "More antiquorum absolute." Opere, I (Part 1), 102. 

uo »p er novam indivisibilium geometriam pluribus modis absoluta." Opere, I (Part 1), 
139. 
111 Proposition V. Opere, I (Part 1), 120-21. 1U Lemma XV. Opere, I (Part 1), 128-29. 



A Century of Anticipation 



125 



Among the eleven demonstrations by the geometry of indivisibles, 
he included one 113 which is— oddly enough— almost identical with the 
mechanical quadrature given by Archimedes in the Method, a work not 
known in the seventeenth century. This coincidence shows how closely 
Cavalieri's geometry of indivisibles resembled the mathematical 
atomism upon which Archimedes' method was probably based. 

Torricelli far outdid his master Cavalieri in the flexibility and per- 
spicuity of his use of the method of indivisibles in making new dis- 
coveries. One of the novel results which pleased him greatly was the 




M 
FIGURE 12 

determination, in 1641, 114 that the volume of an infinitely long solid, 
obtained by revolving about its asymptote a portion of the equi- 
lateral hyperbola, was finite. 116 Torricelli believed that he was the first 
to discover that a figure with infinite dimensions could have a finite 
magnitude; but in this respect he had been anticipated, probably by 
Fermat and Roberval, and certainly in the fourteenth century by 
Oresme. It will be recalled that Oresme, in applying geometric repre- 

113 Proposition XX. Opere, I (Part 1), 160-61. 

114 Bortolottj, "La memoria 'De infinitis hyperbolis' di Torricelli," p. 49. 
U8 "De solido hyperbolico acuto." Opere, I (Part 1), 173-221. 



126 A Century of Anticipation 

sentations to the considerations of Calculator on the latitude of forms 
and infinite series, had shown that a figure with an infinite altitude 
(velocity) could nevertheless have a finite area (distance). Torricelli's 
proof is interesting in that it makes use of the idea of cylindrical in- 
divisibles, whereas those of Cavalieri had invariably been plane. Let 
the hyperbola be rotated about BA, and let ED be a fixed horizontal 
line (fig. 12). Let ACGH be a right circular cylinder with AC as alti- 
tude and A H as the diameter of its base.* Then Torricelli showed that 
for any position of the line NL parallel to ED, the cylinder with alti- 
tude NO and diameter 01 has a lateral surface area equal to the cross- 
sectional area IM of the cylinder ACGH. But the cylindrical surfaces 
NLIO make up the volume of the infinitely long solid of revolution 
FEBDC; and similarly the areas of the circles of diameter IM con- 
stitute the volume of the cylinder ACGH. Therefore the two volumes 
are equal. 116 

The demonstration given by Torricelli shows strikingly the facility 
offered by the comparison of indivisibles, and the extent to which it 
resembles the procedure employed in the integral calculus, in which, 
of course, the cylindrical and the circular elements are given a thick- 
ness, and the limit of the sum, as this thickness approaches zero, is de- 
termined. Torricelli said that he himself was satisfied that the truth 
of this theorem was sufficiently clear; but for the benefit of those not 
so kindly disposed toward indivisibles, 117 he added, as usual, a demon- 
stration by the method of the ancients. 118 

The work of Torricelli is so eminently catholic in its application of 
the ideas and methods suggested by his predecessors and contempo- 
raries that his name on numerous occasions became the center of dis- 
putes concerning priority. Probably no century presents more charges 
of plagiarism than does the seventeenth. This is accounted for largely 
by the fact that the method of indivisibles and related procedures were 
used extensively and effectively by mathematicians of many nation- 
alities, all working on similar problems which were leading toward the 
calculus. Inasmuch as there was no logically established justification 
for these heuristic infinitesimal methods, the interrelations of the di- 
ne "Propterea omnes simul superficies cylindricae, hoc est ipsum solidum acutum EBD, 
una cum cylindro basis FEDC, aequale erit omnibus circulis simul, hoc est cylindro 
ACGH." Opere, I (Part 1), 194. 

117 Ibid. u« Und, t pp . 214-21. 

*Where AH is twice the distance from A to the hyperbola. 



A Century of Anticipation 127 

verse points of view were but vaguely realized, and were frequently 
denied. Added to this was the further difficulty that many contributors 
to the theory of the subject— notably Roberval, Fermat, and Newton 
—either did not publish their results, or did so only very tardily. 
Failure of mathematicians to date their works further added to the 
confusion experienced in attempting to attribute specific contributions 
to individual men. No attempt will be made in this essay to consider 
at length such charges of plagiarism, many of which were not sufficiently 
substantiated. Nevertheless, an effort will be made to make clear, so 
far as is possible in this difficult period, the part each man played in 
the development of the ideas of the calculus. 

It will be recalled that Cavalieri had enunciated in geometrical ter- 
minology what may be considered the first general theorem of the 

n + 1 

calculus: that is, the equivalent of j a Q x n dx = — — -, for all positive in- 

n -f 1 

tegral values of n. The generalization and proof of this theorem for all 
rational values of n (except n = — 1) has been commonly attributed to 
Fermat. 119 However, inasmuch as Fermat did not publish his result 
during his lifetime, it is difficult to determine the relationship between 
his work and the analogous and approximately concurrent results of 
Torricelli and Roberval. It would appear 120 that the enunciation, at 
least, of the generalization is to be attributed to Fermat. The date of 
this is doubtful. It may have been as early as 1635, 121 or as late as 
1643. 122 Whether the demonstrations of the rule given by Roberval for 
positive integers and by Fermat for the general rational case antici- 
pated that given by Torricelli in 1646, in his unorganized treatise, De 
infinitis hyperbolis, is not clear. 123 However, the form of proof employed 

119 See, for example, Zeuthen, Geschichte der Mathematik im XVI. und XVII. Jahrhundert, 
p. 265. 

120 Bortolotti, "La scoperta," p. 215. 

ia See Walker, A Study of the TraitS des Indivisibles of Roberval, pp. 142-64. 

» Bortolotti, "La Scoperta," p. 215. 

m Zeuthen asserts that it is to Fermat that we owe the study of the higher parabolas. 
He says that Fermat was probably in possession of the rule for positive integers in 1636 
and that it is almost certain that he had a general demonstration for all cases in 1644. 
"Notes sur l'histoire des mathematiques," IV. "Sur les quadratures avant le calcul inte- 
gral, et en particulier sur celles de Fermat," (1895), pp. 43 ff. Surico, on the other hand, 
maintains the priority of Torricelli in this connection, placing the discovery in 1641 and 
the generalization by 1646. He concludes that Fermat's quadratures were posterior to 
1654, perhaps about 1656. "L'integrazione di y = x n per n negativo, razionale, diverso 
da — 1 di Evangelista Torricelli." 



128 A Century of Anticipation 

is in each case so peculiar to the ideas of its author that, in considering 
the development of the concepts of the calculus, we may, with- 
out fear of misdirection, consider them as independent of each 
other. 

The demonstration offered by Torricelli is in the manner of Archi- 
medes — purely geometrical and employing the method of exhaustion. 
Given any hyperbola DC (fig. 13), he proved that "the quadrilineum 
EDCF is to the frustum DCBG as the power (dignitas) of BA is to the 
power of AE." m The proof of this he carried out by the use of in- 
scribed and circumscribed figures and the application of several 
lemmas, including the fundamental proposition of Valerio, in which 
it is shown that these figures can be made to differ by less than any 
given area. Torricelli remarked, incidentally, that the same procedure 
and conclusion will also be found to apply, with slight alterations, in 
the case of parabolas. 

This result is equivalent to the analytic statement that if the curve 

is x m y n = k, then the ratio of the areas EDCF and DCBG is — . The 

m 

determination of this ratio is in a general sense equivalent to the 

/b m 
a x~ n dx. In such a rep- 
resentation of the problem in terms of modern symbolism, 125 however, 
there is a strong temptation to read into the author's work the con- 
cepts which are called to mind by the newer notation. One must not 
attribute to Torricelli any such algebraic notion as that implied by the 
modern integral sign. Analytic considerations at no point enter into 
his thought, and there is no indication that he had any desire to es- 
tablish an algorithmic rule of procedure applicable to other cases. The 
result remained a simple geometrical proposition on the ratio of areas, 
although the particular quadrature involved was later to be of funda- 
mental significance in the calculus. 

The danger in interpreting Torricelli 's work in terms of modern no- 
tations and ideas is apparent in an evaluation of his work on tangents. 
Torricelli discovered that if DT is tangent at D to the hyperbola con- 
sidered above (fig. 13), then TE is to EA as the ratio of the powers of 

» Opere, I (Part 2), 256. 

125 Such as that given at some length by Bortolotti, "La memoria." 



A Century of Anticipation 



129 



AB and AE; m that is, if the hyperbola is x m y n = k, the ratio of the 

ft • * 

subtangent to the abscissa is — . An interpretation of this proposition 

m 

has been made, 127 in which it is implied that Torricelli regarded the 

tangent as determined by the secant through C and D as C approaches 

D, and that he was therefore not far from the idea of a differential 

quotient. However, there is nothing in Torricelli's language to justify 

such a conclusion. Torricelli's proof is based, not on the modern idea 

of the tangent as the limit of a variable secant, but upon the ancient 




E 
FIGURE 13 

static definition: a line touching the curve at only one point. He 
showed that the assumption that the line TD, determined by the ratio 
stated above, is not tangent at D— that is, that it intersects the curve 
in another point — leads to a contradiction. There is in this proof, of 
course, nothing of the modern idea of limits. 128 However, in another 
connection Torricelli made use of a dynamic conception of tangents 
which was significantly suggestive in the development of Newton's 
fluxional calculus— that based on the parallelogram of virtual velocities. 



™Opere, I (Part 2), 257. 

m Bortolotti, "La memoria," pp. 143-44. 



«• Cf. also Opere, I (Part 2), 304 ff. 



130 A Century of Anticipation 

The principle of the parallelogram of velocities may be considered 
to be implied 129 in Peripatetic science, but inasmuch as Aristotle did 
not develop the notion of instantaneous velocity, the doctrine long 
failed to be widely employed. Archimedes appears to have applied it 
in geometry, and much later it was again suggested by Leonardo da 
Vinci, 130 and used by Stevin, 131 who, because of a predominant interest 
in statics, thought of it in terms of virtual displacements rather than 
velocities. Upon the clarification by Galileo of trajectories in terms of 
the notion of inertia and of the doctrine of the independence of super- 
imposed effects, the idea of the composition of motions was destined 
to play a significant role in science (particularly in dynamics and 
optics) and mathematics during the seventeenth century. 

Torricelli's determination, by means of the composition of motions, 
of the tangents to parabolas of any positive integral degree furnishes 
a striking illustration of his application of the methods of Galileo and 
Cavalieri. It had been recognized since the fourteenth century that 
the motion of a freely falling body is uniformly accelerated — that is, 
that the velocity increases in proportion to the time elapsed — and 
Galileo had incorporated this fact in his dynamics. It had been stated 
by Calculator, Oresme, and others that the distance covered was con- 
sequently one-half that which would be traversed by a body moving 
uniformly for the same length of time with half of the final velocity of 
the falling body. Galileo reasserted this and added that this implied 
that the distance covered by the falling body varied as the square of 
the time. Torricelli pursued this idea further and inquired what would 
be the state of affairs if the velocity were to vary as the square of the 
time. In this case the distance covered would be given by the sums of 
the squares of the lines in the triangle ABE (fig. 14), where these 
squares represent the velocities of the body for a given time interval 
AB. But Cavalieri had demonstrated that the sum of the squares of 
the lines of the triangle ABE is \ the sum of the squares in the par- 
allelogram ABEL Therefore the distance covered will be \ that which 
would have been covered by a second body moving for the time AB 
with a uniform velocity equal to the final velocity of the first body — 

129 See Duhem, Les Origines de la statique, II, 245. 

130 Duhem, op. cit., II, 245-65, 347-48; cf. also Duhring, Kriliscke Geschichte der allge- 
meinen Principien der Mechanik, p. 15. 

m Lasswitz, Geschkltte der Atomistik, II, 12-13. 



A Century of Anticipation 



131 



or, inversely, the final velocity will be given by three times the dis- 
tance covered. The distance, moreover, will vary as the cube of the 
time. 132 

If, therefore, we imagine a projectile moving with a composite mo- 
tion made up of a uniform horizontal velocity and of a vertical ve- 
locity which varies as the square of the time, the curve traversed will 
be a cubical parabola; if the vertical velocity varies as the cube of the 
time, the curve will be a quartic parabola; and so on. The tangents to 




H 



FIGURE 14 



these curves many now be determined as follows: Let the curve ABC 
(fig. 15) be, for example, the cubic parabola. Then if EB is tangent to 
the curve, we will have ED = 3 AD. This is clear from the fact that 
the moving point will possess at B a double impetus: one horizontal, 
and given by the distance BD: and the other vertical, and given (from 
the considerations above) by three times the vertical distance AD. 
Therefore the direction of the point B, by the composition of these two 
velocities, will be that of the line BE, which is consequently the tan- 
m Opere, I (Part 2), 311. 



132 



A Century of Anticipation 



gent. 133 Torricelli remarked that the same type of argument can like- 
wise be applied to other parabolas, the ratio - — being the degree of 

JUJ 

the parabola. 

Torricelli's method, employing as it does the idea of instantaneous 
direction and implying, therefore, the limit concept, represents a 
marked advance over the stultifying definition of the tangent given 




FIGURE 15 

by the ancient geometers. It indicates, as well, a departure from the 
classical tradition, in the intrusion of the notion of instantaneous ve- 
locity into geometrical demonstrations. Dynamical considerations had 
occasionally forced themselves upon the attention of Greek geometers. 
Archimedes may well have used Torricelli's very method in determin- 
ing the tangent to his spiral; but he did not, any more than his con- 
»/wa., pp. 310-11. 



A Century of Anticipation 133 

temporaries, regard the idea of motion as sufficiently rigorous for 
application in his static and formal geometrical proofs, nor did he even 
develop a science of kinematics. The Scholastic philosophers of the 
fourteenth century, however, had given quantitative representations 
of dynamics, work which was elaborated by Galileo. Torricelli now 
employed these in pure geometry in the determination of tangents to 
parabolas. 

He made use of them also in finding the tangents to a large class of 
curves suggested to him by the work of Archimedes. Torricelli con- 
sidered the curves generated by a point which moves along a uniformly 
rotating line with a velocity which is not necessarily uniform, but is, 
instead, a function of its distance from the fixed point about which the 
line rotates. 134 If the velocity is such that in equal intervals of time the 
distances of the variable point from the fixed point are in continued 
proportion, Torricelli called the curve a "geometric spiral," to dis- 
tinguish it from the "arithmetic spiral" of Archimedes. 135 The equation 
of the geometric spiral may be written in polar coordinates as P = 
ae hd , but Torricelli did not treat these curves analytically. Instead, he 
employed considerations from synthetic geometry and mechanics to 
give him the tangents, as well as the lengths of the curves and the 
areas bounded by them. 

In his use of kinematic representations in mathematics, he may have 
been anticipated by the French mathematicians Roberval 138 and 
Descartes (whom we shall consider shortly) ; but it was largely through 
Torricelli's work that the notion gained such popularity that it was 
accepted by Barrow in his geometry and by the latter 's student, New- 
ton, in the method of fluxions. 

Among the results achieved by Torricelli through the application of 

the methods of exhaustion, of indivisibles, and of the composition of 

motions are to be found a number of remarkable anticipations of those 

found in the calculus. They include, as well as numerous theorems on 

quadratures and tangents, some of the earliest results on rectification. 

Torricelli appears also to have recognized and made use of the fact 

that the problem of tangents was the inverse of that of quadratures. 187 

M "De infinitis spiralibus," Opere, I (Part 2), 349-99. M Ibid., p. 361. 

m See Jacoli, "Evangelista Torricelli ed il metodo delle tangenti detto Meiodo del 
Roberval." 
w Bortolotti, "La memoria," pp. 150-52. 



134 A Century of Anticipation 

He did not, however, attempt to establish upon these methods any 
general rules of procedure which might be applied in all cases. He did 
not regard them as constituting a new type of analysis and conse- 
quently did not seek for a universal algorithm. The composition of 
velocities, for example, could be employed only in the case of curves 
for which the generating motions were known beforehand. Only with 
the analytic methods of Fermat, Descartes, and Barrow, or with the 
calculus of Newton and Leibniz did it become possible to determine 
in general, from the equation of a curve, the motions by which the 
curve could be regarded as traced, or the instantaneous direction. 

Although Torricelli's work marked a significant step toward the 
calculus, the basic concepts employed in it were still far from the 
modern point of view. Torricelli had no more idea of defining the no- 
tion of instantaneous velocity in terms of limits than had his master, 
Galileo — or, for that matter, than his successor, Newton. Similarly, 
although Torricelli was well aware of the fact that the method of in- 
divisibles gave results consonant with those obtained by the methods 
of the ancients, he appears to have been far from realizing that the 
two are to be associated through the limit concept. His view of the 
indivisible resembled strongly the vague mathematical atomism of 
Democritus. Torricelli agreed with the assertion, which he attributed 
to Galileo, 138 that a point is equal to a line; and in "confirming" 
Galileo's demonstration that the area under a velocity-time graph 
represents distance, Torricelli affirmed that in the case of unequal lines 
the number of points on each was the same, but that the points them- 
selves were unequal. 139 The lack of a suitable basis for indivisibles was 
perhaps more serious than the omission of a definition of instantaneous 
velocity, for whereas intuition may serve to clarify the use of velocity 
as an undefined element, there is no such safe guide in the use of in- 
divisibles. Torricelli realized the difficulties involved; but, although he 
wrote a work on paradoxes in the use of infinitesimals, 140 he was unable 
to resolve the logical perplexities. 

The date 1647 is significant in the development of the calculus for 
a number of reasons : in the first place Cavalieri and Torricelli died in 
this year, both at early ages; secondly, Cavalieri's more ambitious 

»* Opart, I (Part 2), 321. 

"• Ibid., p. 259. uo De indivisibilvum doctrina per per am usurpata. Ibid., pp. 415-23. 



A Century of Anticipation 135 

work on indivisibles — the Excrcitationes geometricae sex — appeared; 
thirdly, the ponderous Opus geometricum of the greatest of circle- 
squarers, Gregory of St. Vincent, was published at Antwerp. 

Let us now turn from the development of infinitesimals in Italy, to 
consider briefly a concurrent trend in the Low Countries, of which the 
point of view was somewhat different. Gregory of St. Vincent and 
Cavalieri probably worked independently of each other 141 and were in 
possession of their methods at practically the same time — about 1625 
and 1626 respectively. 142 Both men were directly connected, in their 
mathematical inspiration, with the tradition of Archimedes, and each 
was directly indebted, as well, to others — Cavalieri to Galileo (from 
whom he probably adopted his indivisibles), Valerio, and perhaps 
Kepler; Gregory to Stevin and Valerio, from whom he borrowed the 
idea of giving to propositions involving the method of exhaustion a 
direct and yet rigorous demonstration in place of the Greek reductio 
ad absurdum. 143 Gregory, however, added an element not found in 
their works, for he connected the question with the Scholastic dis- 
cussions on the nature of the continuum and the result of infinite 
division. Archimedes, Stevin, and Valerio had subdivided only until 
the error was less than a certain amount, but Gregory interpreted this 
as meaning an actually infinite subdivision. Instead of the parallelo- 
grams of Stevin and Valerio, he used, in his Opus geometricum (or 
Problemum austriacum, as it is sometimes called), infinitely many 
infinitely thin rectangles; 144 and for the w-sided polygons used by 
Archimedes he substituted an inscribed polygon of an infinite number 
of sides, 145 as had Nicholas of Cusa. 

Gregory applied his conceptions to problems of cubatures by a 
process which he called ductus plant in planum. This phrase referred 
to a means of constructing a geometrical solid from two given plane 
figures. Let the figures be, for example, a semicircle and a rectangle 
having the diameter of the circle as one side (fig. 16). Then, by "ap- 
plying the rectangle to the semicircle," Gregory had in mind the fol- 
lowing: Place the rectangle ABA'B' perpendicular to the plane of the 
semicircle. Then for any point on AB erect in the two figures perpen- 

141 Moritz Cantor, Vorlesungen, II, 818. 

ia Ibid., II, 759; see also Bosnians, "Gregoire de Saint- Vincent." 

la Bosnians, op. til., pp. 250-56. "* Opus geometricum, p. 961. 

144 Wallner, "Ober die Entstehung des Grenzbegriffes." 



136 



A Century of Anticipation 



dicular lines XX' and XZ and upon these complete the rectangle 
XZX'Z' . The geometrical solid, of which this rectangle is a section for 
all positions of X on AB, is the one sought. In this case it is a portion 
of a cylinder, but Gregory applied the process to innumerable other 
figures 146 and found the volumes of the solids obtained. 

It is obvious that in this procedure Gregory of St. Vincent was im- 
plicitly making use of infinitesimals. His view of the nature of these, 
although perhaps less naive than that of Cavalieri, was not clear or 
rigorous. 147 The manner in which he constructed his solid figures would 
suggest that he was thinking in terms of indivisibles. Nevertheless, we 




FIGURE 16 

see in other connections that he did not regard them, as had Cavalieri' 
as being without thickness, but rather in the manner of Kepler, as 
literally making up the geometrical figure. After inscribing in two three- 
dimensional figures very thin parallelepipeds, he added that "these 
parallelepipeds can be so multiplied that they exhaust the body within 
which they are inscribed." 148 

This is perhaps the first use of the word "exhaust" in this sense and 
the earliest example in which a configuration was literally exhausted 
in this manner. In Greek demonstrations by the method of exhaustion 

146 Opus geometricum, Book VII. MT Moritz Cantor, Vorlesungen, II, 818. 

148 "Parallelepipeda ilia ita posse multiplicari ut corpora ipsa, quibus inscribuntur, 
exhauriant." Opus geometricum, p. 739. 



A Century of Anticipation 137 

the figure was thought of simply as approximated, to within a given 
degree of accuracy, by the inscribed or circumscribed figure. The 
Greek method is therefore, in a sense, improperly designated as the 
method of exhaustion; but Gregory apparently used the word in its 
literal sense, allowing the subdivision to continue to infinity. He did 
not explicitly state how one is to visualize the exhaustion of the body 
by means of infinitesimal elements, but he was certainly nearer to the 
modern view than Democritus and the Scholastics, or even Cavalieri 
and Torricelli, had been. Instead of thinking of static indivisibles, he 
reasoned in terms of a varying subdivision, thus approximating the 
method of limits. This fact led Gregory in the direction which was 
ultimately to supply the rigorous basis for the calculus, for his infinite 
subdivision brought him to the notion of the limit of an infinite geo- 
metrical progression. 149 

The Greeks had not attempted to define a curve as the terminus ad 
quern, or limit, approached by the inscribed or circumscribed figure. 
Valerio had made the gap between the method of exhaustion and the 
limit idea narrower by stating directly that the approximating in- 
scribed and circumscribed figures could be made to differ by less than 
any magnitude; and Stevin had aided significantly in this direction 
by his omission, on occasion, of one of the two approximating figures 
and by his use of arithmetic sequences. It was Gregory of St. Vincent, 
however, who gave perhaps the first explicit statement that an infinite 
series defines in itself a magnitude which may be called the limit of 
the series. "The terminus of a progression is the end of the series to 
which the progression does not attain, even if continued to infinity, 
but to which it can approach more closely than by any given inter- 
val." 160 As an illustration, Gregory gave a line segment AK, which 
was subdivided by points B,C, . . . with K as their limit and such that 
the segments AB, BC, . . . were in continued proportion. Gregory, 
however, lost somewhat the force of the limit concept by remarking 
that "the magnitude AK is equal to the magnitude of the whole pro- 
gression . . . continued to infinity; or, which is the same thing, K is 

"» Wallner ("Uber die Entstehung des Grenzbegriffes," pp. 251-52) says this was the 
first use of a truly infinite series: but in this he is mistaken, inasmuch as Calculator, 
Oresme, and other Scholastics had employed them long before, as we have seen, in con- 
nection with the latitude of forms. 

m Opus geometricum, p. 55. 



138 A Century of Anticipation 

the terminus of the ratio AB to BC continued to infinity." 161 In other 
words, AK was not defined as the sum, because K was the limit: 
Gregory pointed out, rather, on the basis of geometrical intuition, that 
the remark that AK is the sum, is equivalent to the statement that K 
is the limit. Nevertheless, he stated more clearly than had anyone pre- 
viously that an infinite series can be strictly considered as having a sum. 

Gregory of St. Vincent recognized also that the paradox in the 
Achilles is to be explained in terms of the limit of an infinite series. 
Assuming, as had the Scholastics and Galileo, that "motion is a kind 
of quantity," he asserted that the speeds of Achilles and the tortoise 
must have a proportion and calculated by geometric progressions at 
what point the positions of the two will coincide. 182 He failed to recog- 
nize in this connection that the question of Zeno was not when or 
where Achilles would overtake the tortoise, but rather how he could do it. 
This appeal in the paradox to sensory experience, instead of reasoning, 
was to be without doubt the chief obstacle to the development of the 
calculus in terms of the limit of an infinite sequence. Although Gregory 
of St. Vincent did not express himself with the rigor and clarity char- 
acteristic of the nineteenth century, his work is to be kept in mind as 
constituting the first attempt explicitly to formulate in a positive 
sense — although still in geometrical terminology — the limit doctrine 
which had been implicitly assumed by both Stevin and Valerio, as 
also, probably, by Archimedes in his method of exhaustion. 

Gregory of St. Vincent maintained that he had squared the circle. 153 

Perhaps it is on account of this fact 154 that it has been said 166 that 

Gregory received only disdain on the part of his contemporaries, his 

memory being rehabilitated by Huygens and Leibniz. 156 On the other 

hand, there can be no doubt that his work exerted a strong influence 

on many of the mathematicians of his time. As a teacher in various 

Jesuit schools, he numbered among his disciples Paul Guldin, Andreas 

Tacquet, 157 Jean-Charles della Faille, 158 and others who used infinites- 

» Ibid., p. 97. 152 Ibid., pp. 101-3. 1M Ibid., pp. 1099 ff. 

154 Bosmans, "Gregoire de Saint- Vincent," pp. 254-55. 

156 Marie, Hisloire des sciences mathematiques, III, 187. 
166 Cf. Leibniz, Mathematische Schriften, V, 331-32. 

157 See Kaestner, Gesckichte der Mathematik, III, 266-84, 442-49, for summaries of 
Tacquet's works. 

188 See Bosmans, "Le Mathematicien anversois Jean-Charles della Faille de la Com- 
pagnie de Jesus." 



A Century of Anticipation 139 

imal considerations, particularly in the then-popular problem of de- 
termining centers of gravity. 

Although Guldin wrote in the old Archimedean manner, his work 
has become well known for two reasons. In the first place, the so-called 
Guldin theorem — that the volume of a solid of revolution is given by 
the product of the rotating area and the distance through which the 
center of gravity moves in one revolution 159 — has been much discussed 
from the point of view of possible plagiarism from Pappus. 160 Secondly, 
Guldin became, as we have seen, the chief critic of the lack of rigor in 
the use of infinitesimals by Kepler and of indivisibles by Cavalieri. 
From the point of view of the development of ideas, however, the work 
of Tacquet was more significant than that of Guldin. 

Andre (or Andreas) Tacquet resembles his contemporary, Torri- 
celli, in the generality of his adoption from his predecessors of varied 
infinitesimal methods. In his Cylindricorum et annularium libri IV he 
gave, for example, four demonstrations of the proposition that the 
volume of a sphere is equal to that of a cylindrical wedge whose base 
is half a great circle of the sphere, and whose altitude is equal to the 
circumference of the sphere. This theorem had been given by a number 
of mathematicians since Kepler, as well as by Archimedes in the 
Method, probably not then extant. Tacquet, however, after proving 
the theorem in two ways by the use of inscribed and circumscribed 
figures, gave two further demonstrations by indivisibles, based on the 
equality of triangles and circular sections. Torricelli had himself been 
satisfied with the rigor of proofs by means of indivisibles, although he 
supplied alternative demonstrations for the benefit of others. Tacquet, 
on the other hand, said that he did not consider that the method of 
Cavalieri was to be admitted as either legitimate or geometrical. 161 He 
maintained that the cylindrical wedge could not, in all strictness, be 
considered as made up of triangles; nor could the sphere be regarded 

168 Kepler had given a special case of this theorem in 1615, in his determination of the 
volume of the tore. 

wo For a bitter discussion on this point, see Smith and Miller, "Was Guldin a plagiarist?" 
The editor of the recent French edition of Pappus' works, Paul Ver Eecke, exonerates 
Guldin in no uncertain terms, saying that the theorem of Guldin cannot be deduced from 
the form in which it is given by Pappus, and indeed that a link by inspiration between the 
two is open to serious doubt. See Ver Eecke, "Le Theoreme dit de Guldin consider^ au 
point de vue historique." 

ltl Cylindricorum et annularium, pp. 23-24; or Opera, 3d pagination, p. 13. 



140 A Century of Anticipation 

as composed of circles. Neither was the wedge generated by the flux 
of a triangle, nor the sphere by the motion of a circle. A geometrical 
magnitude, he asserted, is made up only of homogenea, that is, parts 
of like dimension — a solid of small solids, an area of small areas, 
and a line of small lines — and not of heterogenea, or parts of a lower 
dimension, as Cavalieri had maintained. He therefore felt that a pro- 
posed magnitude was to be exhausted (a word which he undoubtedly 
acquired from Gregory of St. Vincent) by inscribing homogenea within 
them "as in the manner of the ancients." 162 

Tacquet's criticism of the method of indivisibles and his insistence 
on the use, instead, of homogenea was, of course, quite justified. Had 
he tried to reconcile the two points of view by employing the word 
"exhaust" in its literal sense in terms of the doctrine of limits, his 
work might have served to clarify the method of indivisibles which 
men continued to employ, not because they understood its significance, 
but because it invariably gave them correct results. It is the more 
strange that Tacquet did not do this, inasmuch as he developed the 
thoughts of Gregory of St. Vincent on limits. For example, in his 
Arithmeticae theoria et praxis of 1656, he explained the Achilles in 
terms of geometrical progressions; 163 and in another passage he re- 
marked that in a progression continued ad infinitum, in which the 
terms decrease in a given proportion, the smallest term vanishes, 164 
thus applying the criterion of a limit. This arithmetical work of Tac- 
quet appeared almost simultaneously with the Arithmetica infinitorutn 
of John Wallis, in which we shall find the notion of limit more vigor- 
ously proposed. Before reviewing the ideas of Wallis, however, it will 
be necessary to consider at some length the significant anticipations of 
methods of the calculus made by a group of illustrious contemporary 
French mathematicians — Roberval, Pascal, and Fermat — upon at 
least one of whom — Pascal — Tacquet exerted an influence. 

Giles Persone de Roberval held the chair of Ramus at the College 

Royal, a position that depended upon supremacy in an examination 

held every three years, the questions for which were propounded by 

the incumbent at the time. To this fact Roberval attributed the secrecy, 

with respect to his methods and results, which eventuated in his loss 

m Ibid. m Arithmeticae theoria et praxis, pp. 502-3. 

w "Minimus terminus evanescat." Ibid., p. 475; cf. also Bosmans, „1Andr6 Tacquet et 
son traits d'arithm&ique theorique et pratique." 



A Century of Anticipation 141 

to other men of credit for priority. He said that he had considered at- 
tentively the "divine Archimedes," and that from this study he worked 
out for himself "the sublime and never to be sufficiently praised doc- 
trine of the infinite." 166 Roberval seems to have worked out his method 
of indivisibles between the years 1628 and 1634, 166 that is, only a few 
years after Gregory of St. Vincent and Cavalieri had developed theirs 
and before the work of either had been published. The almost simul- 
taneous appearance of these procedures indicates how widespread was 
the tendency toward infinitesimal considerations during the early 
seventeenth century. 

Roberval admitted no inspiration for his work other than that of 
Archimedes. It is quite probable, nevertheless, that he was to some 
extent influenced by Kepler, 167 and portions of his work greatly re- 
semble the ideas of Stevin and Valerio; but we can point to no clear 
indications of indebtedness. 168 It would be interesting to know to just 
what extent the work of Stevin was known in France during the first 
half of the seventeenth century. The similarity of some of the ideas 
expressed by Pascal and Roberval to those found in the writings of 
Stevin half a century earlier is striking; but the former scientists made 
no acknowledgment of indebtedness to the engineer of Bruges, whose 
works had appeared several times in Flemish, Latin, and French. 169 

Roberval was without doubt familiar with the work of Cavalieri, 
which he defended from carping critics; 170 but his view of indivisibles 
appears to have been far less naive than that of the Italian. He said 
quite clearly that in his method he did not regard a surface as really 
composed of lines or a solid as made up of surfaces, but as in actuality 
built up of small pieces of surfaces and solids respectively, these "infi- 
nite things" being regarded "just as if they were indivisibles." 171 In his 
Traiti des indivisibles, Roberval asserted that throughout the discus- 
sion it was to be understood that the phrase "the infinite number of 

1,6 Walker, A Study of the TraUi des Indivisibles of Roberval, pp. 15-16. 

"'Ibid., pp. 142-64. 

u1 Ibid., p. SI. 

"■ Duhem (Les Origines de la statique, I, 290-326) asserts that Roberval knew the work 
of Stevin and Valerio. 

1,9 It has been suggested (see The Physical Treatises of Pascal, p. 4, n.) that Stevin 
was the victim, because of his liberality of thought, of a conspiracy of silence on the part 
of Catholics in the Low Countries. 

lw "Divers ouvrages de M. Personier de Roberval," p. 444. 

171 Walker, A Study of the TraiU des Indivisibles of Roberval, p. 16. 



142 A Century of Anticipation 

points" is used for the infinity of little lines which make up the whole 
line, and that "the infinite number of lines" represents the infinity 
of little surfaces which make up the whole surface, and so on. 172 

Roberval attributed like views to Cavalieri, saying that the latter 
did not really mean that a surface was made up of lines; but in this 
he was, apparently, overgenerous. Influenced principally by the classic 
works of Archimedes, Roberval did not recognize that in Cavalieri's 
work, as in that of Galileo and Torricelli, the atomic and Scholastic 
traditions had operated to modify the author's thought and result in 
the method of indivisibles. Unlike Archimedes, however, Roberval 
substituted the conception of infinity for the method of exhaustion, 
somewhat in the manner of Gregory of St. Vincent, but without ex- 
plicitly formulating the limit concept. He did, however, supply the 
essential element found in our conception of the definite integral, in 
that, after dividing a figure into small sections, he allowed these con- 
tinually to decrease in magnitude, the work being carried out largely 
arithmetically and the result being obtained by summing an infinite 
series. This method, which resembles to a considerable extent the pro- 
cedures of Stevin, contrasts strongly with that of the fixed indivisibles 
of geometric character which is found in the work of Cavalieri. 

It has been indicated that the equivalent of the theorem f%x n dx = 

a H + x 

had been anticipated by Cavalieri for positive integral values 

n + 1 

of n, and by Torricelli for rational values of n (except, of course, for 
n = —1). At about the same time Roberval had arrived at this result, 
perhaps on Fermat's suggestion, through investigations which bring 
out nicely the somewhat different emphasis found in his work. Whereas 
Cavalieri and Torricelli had proceeded on the basis of the purely geo- 
metrical considerations involved in the method of exhaustion and in 
the method of indivisibles, the great French mathematical triumvirate 
of Roberval, Fermat, and Blaise Pascal 173 combined their interest in 
the geometry of Archimedes with an enthusiasm for the theory of 
numbers, and this colored their work. As a consequence, Roberval was 
led to make an association between numbers and geometrical magni- 

m "Divers ouvrages," pp. 249-50. 

173 Their famous contemporaries and countrymen, Desargues and Descartes, in this 
respect display a somewhat different spirit. 



A Century of Anticipation 143 

tudes, which resembles strongly that of the Pythagoreans, particularly 
that of Nicomachus. A line segment, it has been remarked, Roberval 
regarded as made up of an infinite number of little lines, represented 
by points, which can be made to correspond to the positive integers. 
If, now, we consider successively the right isosceles triangles with 
sides made up of 4, 5, 6, . . . points or indivisibles respectively, the 
total number of such units in the triangles will be given as follows: 

The triangle of 4 is 10 = |(4) 2 + *(4) 
The triangle of 5 is 15 = |(5) 2 + 1(5) 
The triangle of 6 is 21 = K6) 2 + *(6) 



The second term on the right in each line is half of the side, and rep- 
resents the excess of the triangle over the half square. This continues 
to diminish, in proportion to the first term, as the number of points and 
lines is increased. Since the number of lines in a geometrical triangle 
or square is infinite, the excess or half of one line, "does not enter into 
consideration." 174 It is therefore clear that the triangle is half the 
square. This argument is evidently roughly equivalent to that indi- 

cated by the notation J^xdx = — . 

Roberval continued this type of work by remarking similarly that 
if the lines should follow one another in the order of the square, the 
sum of all these little lines, or the points which represent them, would 
be to the last, taken an equal number of times, as the pyramid is to 
the prism, that is, as 1 to 3. For example, if we take a square pyramid 
of dots having four on an edge, we have l 2 -+- 2 2 + 3 2 + 4 2 = 30 = 
1(4)' + i(4) 2 + i(4); if there are five on an edge, we have l 2 + 2 2 + 
3 a + 4 2 + 5 2 = 55 = i(5) 3 + K5) 2 + i(5); and so on. In these equa- 
tions the first term on the right is one-third the cube, the second is 
one-half the square, and the last is one-sixth the number of points in 
the edge of the base of the pyramid. Inasmuch as the number of squares 
in a geometrical cube is infinite, the last two terms are as nothing, so 

»4 "Divers ouvrages," pp. 247-48. 



144 



A Century of Anticipation 



that the sum is § the cube. 176 In the same way, the sum of the cubes is 
one-fourth of the fourth power; the sum of the fourth powers is one- 
fifth of the fifth power; the sum of the fifth powers is one-sixth of the 
sixth power, and so on. 176 In other words, Roberval has in this manner 



.» + i 



indicated the equivalent of the theorem, flx n chc = 



»-f 1 



, for positive 



integral values of n. He appears not to have given a demonstration for 
other values of w, 177 as did Torricelli and Fermat. 

The arguments of Roberval resemble the "demonstrations by 

arithmetic" given by Stevin half a century earlier, and similar ones 

of Wallis a few years after Roberval's work. 178 All of these represent 

efforts to express the notion of limit, but Roberval in his work obscured 

B E' F' G' H' V C 



V 



» ' » • P\j 

i » i \ / • 

x x ±--oy.---i 

T — T"J^( \ — r 

Lti^-<^ L ' J ' 

a>^^- } j J r 



2" 
S' 

Q' 



E F G H 

FIGURE 17 



D 



the limit idea somewhat by resorting to his notion of indivisibles. 
Instead of drawing his conclusion from the limits of the arithmetic 
sequences involved, he had recourse, as did most of his contempor- 
aries, to geometrical intuition. Making use of the Pythagorean and 
Nicomachean association of numbers with geometrical points, he re- 
marked that since "the side has no ratio to the cube, . . . adding or 
subtracting a single square has no effect." 179 Intuition of this type led, 

175 Ibid., p. 248. "« Ibid., pp. 248-49. m Zeuthen, "Notes," 1895, p. 43. 

178 Walker (A Study of the Traitt des Indivisibles of Roberval, p. 165) makes no mention 
of Stevin in this respect, incorrectly representing Roberval's work as the first of its type. 
The statement that the idea of an arithmetic limit appeared for the first time in the 
seventeenth century (ibid., p. 35) is, of course, not accurate, inasmuch as Stevin's arith- 
metical work in limits appeared in 1586. 

179 "Divers ouvrages," p. 249. 



A Century of Anticipation 145 

through the neglect of infinitesimals of higher order, to the differential 
calculus of Leibniz, rather than to the method of limits which Newton 
suggested and which triumphed in the end. 

Roberval successfully applied this quasi-arithmetical method of 
indivisibles to varied problems in quadratures. Typical of these is his 
quadrature of the parabola. 180 The procedure which he employed is 
quite different from any given by Torricelli in his twenty-one quad- 
ratures of the parabola. Roberval's method resembles, rather, Stevin's 
demonstration by numbers, reinforced by intuition of indivisibles. Let 
AE = 1, AF = 2, AG = 3, . . . (fig. 17). Then from the definition of 

EL AE? 

the parabola we know that — = , and similarly for the other 

v FM AF 2 

points of division. Hence 

area ADC _ AE(EL + FM + GN + . . .) 
area ABCD ~ AD • DC 

_ A E(AE? + AF* + AG 2 + . . ■) 

AD AD* 
- I 2 + 2 2 + 3 2 + . . . + AD 2 _ x 
AD- AD* ~ 3 

Roberval by similar methods found the areas under other curves, 
such as parabolas of higher degrees, the hyperbola, the cycloid, and 
the sine curve, as well as various volumes and centers of gravity as- 
sociated with these. He may in this connection have anticipated Torri- 
celli in finding the volumes of infinitely long solids. He also used an in- 
genious transformation of one figure into another which came to be 
spoken of as the method of Robervallian lines and which resembled 
the ductus plant in planum of Gregory of St. Vincent. Such transforma- 
tions played a large part in the geometry of the seventeenth century 
because of the lack of a facile method for handling curves whose equa- 
tions involved radicals, but after the development of the calculus these 
lost their popularity, as well as their significance. 

Roberval in his work showed a remarkable flexibility, using divers 
infinitesimal elements, such as triangles, parallelograms, parallele- 
pipeds, cylinders, and concentric cylindrical shells. Throughout it all, 

*>Ibid., pp. 256-59. 



146 A Century of Anticipation 

the idea of limits is implied, but is concealed under the terminology 
of Roberval's method of indivisibles. An adumbration of the method 
of limits is indicated also by the manner in which Roberval reconciled 
the demonstrations by indivisibles with those of the ancient geometers. 
First Roberval showed that the unknown quantity lies between in- 
scribed and circumscribed figures which differ "by less than every 
known quantity proposed." Then he showed that the quantity in 
question bore to the circumscribed figure a ratio less, and to the in- 
scribed figure a ratio more, than the proposed ratio. Finally Roberval 
proved the proposition by the application of a general lemma: "If 
there is a true ratio R : S and two quantities A and B such that for a 
small (quantity) added to A, then this sum has to B a greater ratio 
then R: S and for a small (quantity) subtracted from A, the remainder 
has to B a ratio less than R: S; then I say that A : B as R: S." 181 This 
form of argument, resembling strikingly the corresponding propositions 
of Valerio (with whose work he may have been familiar), is equivalent 
to a statement that the limit of a quotient of variables is equal to the 
quotient of their respective limits. 

In Roberval's propositions on indivisibles one recognizes numerous 
anticipations of the integral calculus, some of which are equivalent to 
the determination of definite integrals of algebraic and trignometric 
functions. Roberval was concerned, as well, with problems of the differ- 
ential calculus — for he developed a method of tangents so much like 
that of Torricelli that charges of plagiarism arose. 182 He regarded every 
curve as the path of a moving point, and accepted as an axiom that 
the direction of motion is also that of the tangent. 183 By looking upon 
the motion of the point as made up of two component movements, 
he found the tangent by determining the resultant of these. Thus he 
found the tangent to the parabola by making use of the fact that, 
since this curve is the locus of points equidistant from the focus and 
the directrix, it may be regarded as generated by a point moving with 
a compound motion made up of a uniform motion of translation away 
from the directrix and an equal uniform radial motion away from the 
focus. By the parallelogram (in this case a rhombus) of velocities it is 

181 Walker, A Study of the Traiti des Indivisibles of Roberval, pp. 38-39. 
m "Divers ouvrages," pp. 436-78; Walker, op. cit., pp. 142-64. Moritz Cantor (Vorle- 
sungen, II, 808-14) concludes that the charges are not substantiated. 
ms "Divers ouvrages," p. 24. 



A Century of Anticipation 147 

therefore determined that the resultant velocity — and consequently 
the tangent to the parabola at any point— will be in the direction of 
the bisector of the angle between the focal radius at the point and the 
perpendicular from the point to the directrix. This direction being 
known, the tangent can be drawn. 184 The motions involved here are 
different from those used for the same curve by Torricelli, but the 
underlying idea of the composition of movements is essentially the 
same. The method is, of course, subject to the difficulty that one must 
in some way discover the laws of motion before one can determine the 
tangent. Roberval appears, from his correspondence with Fermat in 
1636, to have had another method of tangents which proceeded ana- 
lytically and which he says was connected with the problem of quad- 
ratures. 186 This might have been significant in the history of the 
calculus, but it apparently was lost. 

It is difficult to determine the extent of Roberval!s influence on con- 
temporary mathematicians inasmuch as his Traite des indivisibles was 
not published until 1693 — that is, only after the calculus itself had 
been made known. It is very probable, however, that he had a strong 
influence upon Pascal the younger, whose father, Etienne Pascal, was 
a close friend of Roberval. 

Blaise Pascal in a sense represents the highest development of the 
method of infinitesimals carried out under the traditions of classical 
geometry. He was not so much a creative genius as a mathematician, 
scientist, and philosopher, with a remarkable flair for clarifying ideas 
which had been somewhat vaguely set forth by others, and for sup- 
plying these with a reasonable basis. 186 This penchant of Pascal's is 
well illustrated in science by his lucid organization of the principles of 
hydrostatics; in mathematics one sees it in his exposition of the nature 
of infinitesimals, in which, to be sure, one perceives also a touch of his 
characteristic mystical turn. 

Pascal was not a professional geometer, and as a result his geo- 
metrical work was accomplished in two periods which were separated 
by an interval of mathematical inactivity (from 1654 to 1658) during 
which he devoted his interests to theology. These two periods, more- 
over, are characterized by somewhat different views as to the nature 

184 Ibid., pp. 24r-26. » See Moritz Cantor, Vorlesungen, II, 812. 

m Bosnians, "Sur l'oeuvre mathematique de Blaise Pascal.' 5 



148 A Century of Anticipation 

of infinitesimals. Pascal had two predominating interests in mathe- 
matics: geometry and the theory of numbers. Toward the end of the 
first of his two periods of,mathematical work the latter was dominant, 
and at this time he applied the theory of infinitesimals to his work on 
the arithmetic triangle. Although this is usually called Pascal's tri- 
angle, the ordering of binomial coefficients had been known to Stifel 
long before. 187 

In this connection he enunciated, in the Potestatum numericarum 
summa of 1654, the theorem on the integral of x n , which we have met 
with in the work of Cavalieri, Torricelli, and Roberval. Pascal's dem- 
onstration of this 188 is derived not from classic geometrical propositions 
alone, but from an examination of the figurate numbers represented in 
the arithmetic triangle — a form of proof suggestive of that of Roberval, 
which appears not to have been generally known at Paris at the time. 189 
In the arithmetic triangle the numbers in the first row (or column) may 
be considered units or points making up a line. Those in the second 
row represent the sums of the numbers in the first row and may be 

11111.. 

12 3 4.. 

13 6.. 
14.. 

1 . . 

considered therefore as the sums of points or units — that is, as lines. 
The numbers in the third row, which are in turn the sums of those in 
the second row, may therefore be considered as the sums of lines, that 
is, as triangles. The numbers in the fourth row similarly represent 
pyramids. Geometrical intuition now fails, but one can continue by 
analogy. 190 

From such geometrical considerations and from the numerical re- 
lationships within the triangle, Pascal was led immediately to con- 
sider the sums of powers of the positive integers. Recalling the results 
by the geometrical procedures of the ancients for the sums of the 

187 See Bosmans, "Note historique sur le triangle arithme'tique de Pascal." 
m (Euvres, III, 346-67, 433-593. "» Zeuthen, "Notes," 1895, p. 43. 

w Cf. Bosmans, "Sur l'interpr6tation geome'trique, donnee par Pascal a l'espace a 
quatre dimensions." 



A Century of Anticipation 149 

squares and of the cubes, he recognized that these were not imme- 
diately applicable to powers of higher degree. Pascal, however, de- 
veloped a general arithmetical method for determining the sum, not 
only in the case of terms which are integral powers (of the same de- 
gree) of the first N natural numbers, but also for powers (of the same 
degree) of any integers in arithmetic progression. Pascal expressed his 
result rhetorically, but this may be given in symbolic form by the 
equation: 
■ + , Ci<te w + " + 1 C«#Z ( — I) + . . . + n + 1 C„J w S (1) = (a + Nd) n + l 

-a" + l - NdT + l , 
where a is the first term of the progression, d the common difference, 
N the number of terms, n the degree of the powers in question, " + ! Q 
the number in the (i + l)st column and the (n — i + 2)nd row in 
Pascal's triangle, and S 0) the sum of the jth powers of the terms of the 
progression. 

As Pascal remarked, it will be obvious to any one who is at all fa- 
miliar with the doctrine of indivisibles that this result can be applied 
to the determination of curvilinear areas. In order to find the area 
under the curve y = x n , for example, the surface in question is to be 
regarded as the sum of ordinates which are the nth powers of abscissae 
chosen in arithmetic progression (with first term zero and with common 
difference equal to unity), of which in this case there will be an infinite 
number. Moreover, a single point adds nothing to the length of a line: 
nor does the addition of a line to a surface cause any difference in area, 
for the former is an indivisible with respect to the latter. Or, speaking 
arithmetically, roots do not figure in a ratio of squares, nor squares in 
a ratio of cubes, and so on. 191 The rule given above consequently be- 
comes, on calling the greatest abscissa b and on neglecting as zero the 
terms of lower order, {n + l)S (w) = b n + \ In a general sense this is, 

b n + l 

of course, the equivalent of the expression $&*dx = . 

n-f 1 

Or, translating this into the terminology often used at the time, the sum 
of lines in a triangle is half the square of the longest; the sum of the 
squares of the lines is one- third the cube; the sum of the cubes is one- 
fourth the "square-square," and so on. 

The essential point in Pascal's demonstration is the omission of 
*» (Euvres, HI, 366-67. 



150 A Century of Anticipation 

terms of lower dimension. This type of argument has frequently been 
attributed to Cavalieri, 192 but there appears to be no basis for such a 
view. Cavalieri 's method was based upon a strict correspondence of the 
indivisibles in two figures, and there were no unpaired or omitted ele- 
ments. The method of dropping terms seems to have entered in the 
work of Roberval and Pascal through the association of the indivisible 
of geometry with arithmetic and the theory of numbers. The geo- 
metrical intuition of indivisibles of lower dimension was, in their work, 
carried over into arithmetic to justify the neglect of certain terms of 
lower degree. Pascal went so far as to compare the indivisible of ge- 
ometry with the zero of arithmetic, much as Euler later regarded the 
differentials of the calculus as nothing but zeros. 

This neglect of quantities, as found in Pascal, has been character- 
ized 193 as the basic principle of the differential calculus. Such a desig- 
nation is indeed misleading, for the subject is no longer explained in 
terms of the omission of fixed infinitesimals. Nevertheless, the work of 
Pascal exerted perhaps the strongest influence in shaping the views 
of Leibniz, who adopted into his calculus as fundamental the doctrine 
that "differences" of higher order could be neglected. Newton also oc- 
casionally lapsed into this type of argument in dropping out of the cal- 
culation "moments" which did not add significantly to the result. For 
almost two centuries mathematicians tried to justify such procedures, 
but in the end the basis of analysis was found, not in these, but rather 
in the method of limits toward which the geometrical method of ex- 
haustion and arithmetical modifications of this by Stevin, Tacquet, 
Roberval, and others had pointed. 

In answering the objections of those of his contemporaries who held 
that the omission of infinitely small quantities constituted a violation 
of common sense, Pascal had recourse to a favorite theme — that the 
heart intervenes to make this work clear. In this case what is necessary 
is the "esprit de finesse," or intuition, rather than the "esprit de geo- 
metric," or logical thought, much as the action of grace, as well as 
physical experience, is above reason. In this respect the paradoxes of 
geometry are to be compared to the apparent absurdities of Christi- 

m Ball, History of Mathematics, p. 249; Cajori, History of Mathematics, p. 161; Marie, 
Histoire des sciences mathematiques, IV, 72; Milhaud, "Note sur les origines du calcul 
infinitesimal," p. 35. 

"• Simon, "Zur Geschichte und Philosophic der Differentialrechnung," p. 120. 



A Century of Anticipation 151 

anity, the indivisible being to geometrical configurations as our justice 
to God's. 194 

The mysticism which Pascal often displayed in his attitude toward 
the infinitesimal does not appear in all of his work. Particularly in the 
later period of his mathematical activity — in which his interest cen- 
tered about the cycloid, the curve which Montucla called "la pomme 
de discorde" 195 because it engendered so many quarrels with respect to 
priority — his view appears to be modified. In connection with problems 
such as those in his Traite des sinus du quart de cercle of 1659 in which 
he balanced elements as Archimedes had done in his mechanical 
method, he used the language of infinitesimals in speaking of the sum 
of all the ordinates; but he added that one need not fear to do this, 
inasmuch as what is really meant is the sum of arbitrarily small rec- 
tangles. 196 In his later numerical demonstrations also Pascal sought to 
avoid arguments based upon the neglect of infinitely small quantities. 
The Aristotelian view had denied the existence in the realm of number 
of the infinitely small, while admitting, as a potentiality, the infinitely 
great. Pascal, on the contrary, maintained in De Vesprit geometrique 
that in the sphere of number the infinitely great and small are com- 
plementary. Corresponding to every large number, such as 100,000, 
there existed a small one, the reciprocal i o*ooo so that the existence 
of the indefinitely large implied that of the indefinitely small. Number, 
he held, was as much subject to the two infinities — in greatness and in 
smallness — as were such other undefined primitive terms in geometry 
as time, motion, and space. 197 The contrast between discrete and con- 
tinuous magnitudes was not so great as Aristole had felt, and was, in 
fact, vanishing with the spread of analytic methods in geometry. 

The change in Pascal to a clear point of view with respect to infini- 
tesimals may have been the result of friendship with Roberval, who had 
said that Cavalieri did not really think of indivisibles as lines. It may 
equally well have come from Pascal's reading of Tacquet's Cylindri- 
corum et annularium, 19 * in which the author denied the validity of con- 
cluding anything about the ratio of surfaces from the ratio of their 
indivisibles, or lines. Tacquet had been particularly emphatic in deny- 

m See (Euvres, XII, 9, XIII, 141-55. 

m Montucla, Histoire des mathematiques, II, 52. 

"• (Euvres, IX, 60-76. IW (Euvres, IX, 247, 253, 256, 268. 

i» Bosnians, "La Notion des indivisibles chez Blaise Pascal." 



152 A Century of Anticipation 

ing that a configuration could be thought of as composed of heterogenea, 
or elements of a lower dimensionality. Pascal was in general agreement 
with him on the question of homogeneity, but his view of the transition 
from the finite to the infinite was different. Tacquet was inclined to- 
ward the limit idea of Gregory of St. Vincent, although he preferred 
to avoid the difficulty by returning to the clarity afforded by the 
method of exhaustion. 

Pascal, on the other hand, looked upon the infinitely large and the 
infinitely small as mysteries — something which nature has proposed 
to man, not to understand, but to admire. 199 Furthermore, Tacquet 
had made use of the limits of infinite series, as had also Stevin and 
Roberval. The work of Pascal, however, was carried out in connection 
with the older theory of numbers and classical geometry of which he 
considered his method an elaboration. The newer analytic procedures 
of Fermat and Descartes did not appeal to him, and for them he sub- 
stituted a remarkable facility in the manipulation of geometric trans- 
formations, similar to those of Gregory of St. Vincent and Roberval. 
Through these he related the figurate sums of his number theory to 
problems in the synthetic geometry of continuous magnitudes and an- 
ticipated numerous results of the integral calculus, including the 
equivalent of integration by parts. His underestimation of the value 
of the algebraic and the analytic viewpoints may have been responsible 
not only for his inability to define the central and unifying concept of 
the integral calculus — that of a limit of a sum — but also for his failure 
to recognize the inverse nature of the problems of quadratures and 
tangents. 

The idea and figure of what is now called the differential triangle had 
appeared on several occasions before the time of Pascal, and even as 
early as 1624. Snell, in his Tiphys Batavus, had thought of a small 
spherical surface bounded by a loxodrome, a circle of latitude, and a 
meridian of longitude as equivalent to a plane right triangle. 200 Numer- 
ous diagrams somewhat resembling the differential triangle are to be 
found in the geometrical works of the middle seventeenth century, 
such as those in the De infinitis hyperbolis of Torricelli and the Traite 
des indivisibles of Roberval, with which Pascal may have been familiar. 

"» (Euvres, DC, 268. 

100 Aubry, "Sur Phistoire du calcul infinit6simal entre les ann6es 1620 et 1660," p. 84. 



A Century of Anticipation 



153 



In all of these, however, the significance of the quotient of two sides of 
the triangle for the determination of tangents appears to have escaped 
emphasis. It was much the same with Pascal. In connection with a 
diagram (see fig. 18) from his Traite des sinus du quart de cercle of 1659, 
he remarked that AD is to DI as EE is to RR or EK, and that for 
small intervals the arc may be substituted for the tangent. Pascal made 
use of these lemmas to determine the sum of the sines (ordinates) of a 
portion of the curve, that is, the area under this portion. If Pascal had 
at this point only been more interested in arithmetic considerations 
and in the problem of tangents, he might have anticipated the im- 
portant concept of the limit of a quotient and have discovered the 




R I R 
FIGURE 18 



significance of this for the determination of both tangents and quad- 
ratures. Had he done this, he would have hit upon the crucial point in 
the calculus some seven years before Newton and about fourteen years 
before Leibniz. The latter, who later, as we shall see, made use of this 
very diagram to establish his infinitesimal calculus, said, in 1703, in a 
letter to James Bernoulli, that sometimes Pascal seemed to have had 
a bandage over his eyes. 201 This apparent lack of imagination was very 
likely the result of a predilection for the classical, such as later re- 
strained the scientist Huygens also from making full use of the new 
procedures. 

m See Leibniz, The Early Mathematical Manuscripts, pp. 15-16, and Mathematische 
Schrijtm, HI, 72-73, n. 



154 A Century of Anticipation 

Pierre de Fermat, the friend of Pascal and perhaps the greatest 
French mathematician of the century, possessed a singular erudition 
and displayed an enthusiastic interest in Greek and Latin philology. 
This led him to study carefully such classic mathematical works as 
those of Archimedes, Apollonius, and Diophantus. The influence of 
the first of these three had been very strong for almost two centuries, 
but in the work of Viete and of Fermat we have evidence of the im- 
pression made by the other two also, as well as by the Arabic and 
Italian development of algebra. Viete realized the facility to be gained 
in the handling of geometric problems by their reduction to the solu- 
tion of algebraic equations, a procedure which he therefore followed 
whenever possible. 202 Viete's equations betrayed their origin in ge- 
ometry, in that he was always careful to have them all homogeneous; 
but his work was nevertheless, in a sense, an inversion of the Greek 
view, in accordance with which algebraic equations were reduced to 
geometric constructions for purposes of solution. 

Fermat was familiar with the methods of Viete and developed them 
into an analytic geometry at about the time that Descartes was pre- 
paring his famous Geometrie of 1637. The work of Fermat and Des- 
cartes went much further than did either the algebraic solution of 
geometric problems by Viete or the graphical representation of vari- 
ables by Oresme, for it associated with each curve an equation in 
which are implied all of the properties of the curve. This recognition, 
which Fermat expressed in calling the equation the "specific property" 
of the curve, constitutes the basic discovery of analytic geometry. 
Although in publication Fermat was anticipated by Descartes, he far 
outdid his rival in the application of the new point of view to the 
problems of infinitesimal analysis, which the books of Kepler and 
Cavalieri had popularized. 

All of the anticipations of methods of the calculus which we have so 
far considered were related to geometry. Infinite series had sometimes 
been employed, but they were derived from the geometrical represen- 
tation of the problem. Infinitesimal lines, surfaces, and solids had been 
used, but not infinitesimal numbers. Aristotle had denied the infinitely 
small in arithmetic, for the obvious reason that since number was a 

202 For an unusually extensive account of this work, see Marie, Histoire des sciences 
matktmotiques, III, 27-65. 



A Century of Anticipation 155 

collection of unities, no number could be smaller than one. As a result 
of the algebra and the analytic geometry of the sixteenth and seven- 
teenth centuries, this attitude had been modified, as has already been 
seen in the case of Pascal. The present-day view of the symbols enter- 
ing into an equation is that they represent, in general, continuous 
variables; for Fermat and Descartes, however, they represented in- 
determinate constants 203 to which line segments could be associated, 
the tacit assumption being made that to every segment there corre- 
sponded some number. To such a view there was nothing incongruous 
in the idea of infinitesimal constants or numbers, since they would 
correspond to the geometrical infinitesimals which were being used so 
successfully. These numerical infinitesimals arose first through some 
interesting problems considered by Fermat. 

Pappus had spoken of a "minima et singularis proportio," and this 
led Fermat to dwell on the fact, as he explained in a letter of 1643, 
that in a problem which in general has two solutions, the maximum or 
minimum value gives but a single solution. 204 Thus if a line of length 
a is divided by a point P into two parts, x and a — x, there are in 
general two positions of P which will make the area of the rectangle on 
x and a — x a given quantity, A. For the maximum area, however, 
there is only one position, the mid-point. 

From this fact Fermat was led to formulate his remarkably ingenious 
and fruitful method for determining maximum and minimum values. 
His method first appeared in an article in 1638; but Fermat said that 
the discovery went back some eight or ten years previous. 205 The argu- 
ment in this is as follows: Given a line segment of length a, mark off 
from one end a distance x. The area on the segments x and a — x will 
then be A = x(a — x). If instead of the distance x one were to mark 
off the distance x + E, the area would be A = (x + E) (a — x — E). 
For the maximum area the two values will be the same, from Pappus' 
observation, and the points x and x -f- E will coincide. Consequently, 

203 Wallner, "Entwickelungsgeschichtliche Momente bei Entstehung der Infmitesimal- 
rechnung," p. 119. 

204 See Giovannozzi, "Pierre Fermat. Una lettera inedita." 

205 See Wieleitner, "Bermerkungen zu Fermats Methode der Aufsuchung von Extremen- 
werten." In a letter to Roberval, written in 1636, Fermat said that in 1629 he was in pos- 
session of his method of maxima and minima. See Paul Tannery, "Sur la date des princi- 
pales d6couvertes de Fermat"; cf. also Henry, "Recherches sur les manuscrits de Pierre 
de Fermat." 



156 A Century of Anticipation 

setting the two values of A equal to each other and letting E vanish, 

. . . a 

the result is x = -. z06 
2 

The procedure which Fermat here employed is almost precisely that 
now given in the differential calculus, except that the symbol Ax (or 
occasionally h) is substituted for E. In his work there appeared, for 
perhaps the first time, the idea which has become basic in such prob- 
lems — that of changing the variable slightly and then letting this 
change vanish. However, the reasoning by which Fermat supported 
his method is far less clear than that given at the present time. Modern 
analysis makes use of the concept of the limit, as the change Ax ap- 
proaches zero. Fermat, however, seems to have interpreted the opera- 
tion as one in which E vanishes in the sense of actually being zero. For 
this reason, as Berkeley remarked in the following century, it is diffi- 
cult to see by what right he took the positions x and x + E to be dif- 
ferent and yet in the end said that they coincide. Fermat's argument 
has frequently been interpreted in terms of the limit concept, 207 in 
which E is to be regarded as a variable quantity approaching zero. 
Fermat, however, does not appear to have thought of it in this way. 208 
In fact the function concept and the idea of symbols as representing 
variables does not seem to enter into the work of any mathematician 
of the time. 

In answering criticisms of his method, Fermat presented a statement 
of his reasoning which appears to link it with the remarks of Oresme 
and Kepler on the change at a maximum point. 209 He justified the 
equating of the two values of A by remarking that at a maximum 
point they are not really equal but they should be equal. He therefore 
formed the pseudo-equality 210 which became equality on letting E be 
zero. From this it is clear that he was thinking in terms of equations 
and the infinitely small, rather than of functions and the limit concept. 

"• Fermat, (Euvres, I, 133-34, 147-51; III, 121-22; Supplement, pp. 120-25. Cf. also 
Voss, "Calcul differentiel," p. 246. 

m Duhamel, "Memoire sur la m^thode des maxima et minima de Fermat, et sur les 
methodes des tangentes de Fermat et Descartes"; cf. also Mansion, "M6thode, dite de 
Fermat, pour la recherche des maxima et minima.' ' 

108 See Wallner, "Entwickelungsgeschichtliche Momente," pp. 122-23; cf. also Paul 
Tannery, Notions kistoriques, p. 344. 

** Paul Tannery, however, thinks Fermat borrowed nothing from Kepler, whose works 
he probably had not read. See his Review of Vivanti, II concetto d'infinitesimo, p. 232. 

210 "Adaequalitas." See (Euvres, I, 133-79, for his justification. 



A Century of Anticipation 



157 



None the less, the method worked so beautifully that it found a ready 
acceptance among mathematicians. As a result, infinitesimals were un- 
critically introduced into analysis, to become firmly intrenched as the 
basis of the subject for about two centuries before giving way, as the 
fundamental concept of the calculus, to the rigorously defined notion 
of the derivative. Even now the subject is generally known as the 
"infinitesimal calculus," in spite of the fact that the infinitesimal, while 
of great pragmatic value in adding to the facility of manipulation of 
the subject in exercises, is logically secondary and even unnecessary. 

Fermat was led by the success of his method to apply it, about 1636, 
to the determination of tangents to curves. This he did as follows: Let 
the curve be a parabola (fig. 19). Then from the "specific property" of 




FIGURE 19 
the curve it is clear that if we set OQ = a, VQ = b, and QQ' = E, we 



shall have 



> 



Torricelli, in his work on the tangents to 



b + E (a + E)* 
parabolas, had frequently set down such inequalities. 211 However, 
whereas Torricelli had made use of arguments by a reductio ad ab- 
surdum, Fermat's characteristic procedure resembles more closely the 
method of limiting values. Inasmuch as for small values of E the point 
P' may be regarded as practically on the curve as well as on the tan- 
gent line, the inequality becomes, as in the method for maximum 
values, a pseudo-equality. By allowing E to vanish, this pseudo- 
equality becomes a true equality and gives the desired result, a = 
2b. m 
** See Opere, I (Part 2), 304 ff., 315 ff. m (Euvres, I, 134-36; III, 122-23. 



158 A Century of Anticipation 

The method of tangents Fermat believed to be an application of his 
method for maxima, but he was unable to explain what quantity he 
was maximizing. Descartes naturally supposed that it was the length 
of the line from the curve to a fixed point O on the axis of the parabola. 
However, on carrying out Fermat's method on this assumption, the 
result he obtained was, of course, different from that of Fermat. What 
Descartes had really found was the normal to the curve — that is, the 
minimum distance from a point on the axis to the curve. This would 
have furnished an excellent illustration of the method for determining 
a maximum or a minimum value, but inasmuch as Fermat had not 
given a rule for distinguishing maxima from minima, neither he nor 
Descartes recognized it as such. Descartes simply concluded that al- 
though the result Fermat had obtained was correct, the method was 
not generally applicable. 

Perhaps as the result of the unnecessarily bitter criticism and super- 
cilious attitude of Descartes, Fermat later modified his explanation for 
determining tangents. 213 Instead of interpreting the method in terms 
of maxima, he said that the point P' was indifferently taken as on the 
curve or on the tangent line. Then after forming the pseudo-equality, 
the quantity E was to vanish, to give the desired result. This procedure 
is strictly comparable to that now employed in the calculus, the theo- 
retical justification of which is given in terms of limits; but the ex- 
planation of Fermat resembles rather the neglect of infinitesimals to 
be found in the work of Leibniz. It is also strikingly suggestive of the 
doctrine of perfect and imperfect equations, presented almost two 
hundred years later by Carnot in his attempted concordance of the 
conflicting views of the calculus then prevailing. 

Fermat applied analogous considerations to the problem of deter- 
mining the center of gravity of a segment of a paraboloid, again under 
the misapprehension that he was employing the method of maxima 
and minima. In this he let the center of gravity, O, of the segment be 
a units from the vertex. On decreasing the altitude, h, of the segment 
by E, the center of gravity is changed. Fermat, however, knew from 
a number of lemmas that the distances of the centers of gravity of the 
two segments are proportional to the altitudes and that the volumes 
of the segments are to each other as the squares of the altitudes. By 
* u Duhamel, "M6moire sur la m6thode des maxima et minima de Fermat/ 9 pp. 310-16. 



A Century of Anticipation 159 

taking moments about O, he was able to make use of these facts to set 
up a pseudo-equality involving a, h, and E. In accordance with his 
general principle, he allowed E to vanish and obtained the result a — 

p. 214 

The determination of the center of gravity of the paraboloidal seg- 
ment did not constitute a new result. Archimedes had calculated this 
some nineteen hundred years earlier in the Method, and Commandino 
and Maurolycus had rediscovered it only a century before. Neverthe- 
less, this exercise of Fermat is significant in the history of the calculus 
as the first determination of the center of gravity by means of methods 
equivalent to those of the differential calculus, instead of by means of 
a summation resembling those of the integral calculus. Fermat's friend 
Roberval was astonished that one should be able to obtain by means of 
this maximum and minimum method a result which had generally been 
derived from summation considerations. The integral calculus is, of 
course, implied in the lemmas which Fermat employed in this con- 
nection, and the method of maxima served somewhat indirectly to 
determine simply the value of the constant of proportionality entering 
into these. Nevertheless, this theorem might have led Fermat to a 
recognition of the significance of the inverse nature of summation and 
tangent problems. That this escaped him is the more strange, in that 
he developed remarkable procedures for the determination of quad- 
ratures as well as for tangents. 

a n + i 

The equivalents of what we express as fffidx = had appeared 

n + 1 

in various forms in the work of Cavalieri, Torricelli, Roberval, and 
Pascal. Fermat also gave demonstrations of this rule — in fact, he may 
well have anticipated all the others in this respect — one of which is 
strikingly different from those given earlier. 216 In his earlier investiga- 
tions, about 1636, he appears to have made use of the inequalities 

HI J. i 

1"» + 2 m + 3 m + ...+«"* > > r + 2 m + 3 M + ... + (»- l) w 

m + 1 

to establish the result for positive integral values of n. This constituted 

a generalization of the inequalities of Archimedes which was known also 

» (Euvres, I, 136-39; III, 124-26. 

** For an excellent exposition of this method of quadratures, see Zeuthen, "Notes," 
1895, pp. 37-80. 



160 A Century of Anticipation 

to Roberval. Fermat may also have given a proof, based upon the for- 
mation of figurate numbers, similar to that of Pascal. 216 However, be- 
fore 1644 he had found the quadratures, cubatures, and centers of 
gravity of the "parabolas" of fractional degree a m y n = b n x m , n7 curves 
which he seems to have been the first to propose, but which were in- 
vestigated also by Cavalieri, Torricelli, Roberval, and Pascal. Fermat, 
therefore, probably possessed as early as that date a general proof of 
the theorem for rational fractional values as well, although the defin- 
itive redaction of this was not made until 1657. 218 

In this connection, Fermat's procedure constitutes a generalization 
of one found in the Opus geometricum of Gregory of St. Vincent, al- 
though Fermat may not have known of this work, for he here mentions 
only Archimedes. Gregory had shown that if along the horizontal 
asymptote of a rectangular hyperbola points are marked off whose 
distances from the center are in continued proportion, and if at these 
points ordinates are erected to the curve, then the areas intercepted 
between these are equal. 219 Fermat modified this process in such a way 

that it could be applied to both the general fractional hyperbolas and 

p 

parabolas. In finding the area under y = x 9 , 220 for example, from to 
x, he would take points on the axis with abscissas x, ex, e 2 x, . . . where 
e < 1 (fig. 20). Then erecting the ordinates at these points, the areas of 
the rectangles constructed upon successive ordinates will form an in- 
finite geometric progression. As Gregory of St. Vincent and Tacquet 
had earlier found the sums of such progressions, so here Fermat de- 

t±il \ - e \ 
termined the sum of the rectangles as x 9 j p + q J. In order 

v - « ' I 

to find the area under the curve, however, one must have not only an 
infinite number of such rectangles, but the area of each must be in- 
finitely small. This can be brought about by setting e = 1. Before 

218 Ibid., pp. 42-43. 

217 Mersenne, Cogitata physico-mathematica; see preface to Tractatus meckanicus. See also 
(Euvres de Fermat, I, 195-98. 
a8 Zeuthen, "Notes," 1895, pp. 44 ff. 

219 Opus geometricum, Proposition CIX, p. 586. 

220 The notation of Fermat has been here slightly modified to make the meaning more 
clear. The equations of Fermat retained the homogeneous character found in Viete. 



A Century of Anticipation 



161 



doing this, however, Fermat made the substitution e = E? in order to 
evaluate the indeterminate form. The sum then becomes 



t±s(\ _ & 



1 - E p 



+ 9 



P+9 



= X 



(1 -£)(! +£ + £* + ... +£ g " 1 ) 

(1 - £)(1 + £ + & + • • • + E p+q ~ l ) 

p_±j 

gx 9 



When e approaches 1, E does likewise, and the sum is then , 

P +9 
which is the area under the curve. By taking e > 1, Fermat applied 
the same method to the fractional hyperbolas also, finding the area 
under these from any abscissa to infinity. 221 




FIGURE 20 

In these quadratures we see most of the essential aspects of the defi- 
nite integral — the division of the area under the curve into small ele- 
ments of area, the approximate numerical determination of the sum of 
these by means of rectangles and the analytic equation of the curve, 
and finally an attempt by Fermat to express the equivalent of what 
we would call the limit of this sum, as the number of elements is in- 
definitely increased and as the area of each becomes indefinitely small. 
One is almost tempted to say that Fermat recognized all the aspects 
except that of the integral itself; that is, he did not recognize the op- 
eration involved as significant in itself. The procedure was for him, as 

» (Euvres, I, 255-88; III, 216-40. 



162 A Century of Anticipation 

it had been for all of his predecessors, simply that of finding a quad- 
rature — of answering a specific geometrical question. Only with 
Newton and Leibniz were the processes involved in infinitesimal con- 
siderations recognized as constituting operations, independent of any 
geometrical or physical considerations, to which characteristic names 
were applied. 222 

That a curvilinear area, such as those under Fermat's parabolas and 
hyperbolas, could be equal to one bounded only by straight lines had 
been known in antiquity. It had long been held impossible, however, 
that a curved line could be exactly equal in length to a straight line, 223 
and Fermat shared this view with a number of his contemporaries. 
Sluse and Pascal in this connection expressed admiration for the order 
of nature, which refused to allow a curve to equal a line. 224 Gregory of 
St. Vincent, Torricelli, Roberval, and Pascal, nevertheless, had by 
infinitesimal and kinematic means compared the arcs of spirals with 
those of parabolas. Then, shortly before 1660, there suddenly appeared 
a number of rectifications of curved lines by William Neil, Christopher 
Wren, Heinrich van Heuraet, John Wallis, and others. 226 

These were, in general, based upon approximations to the curve by 
means of polygons, followed by applications of infinitesimal or limit 
considerations. Upon hearing of them, Fermat himself carried out a 
rectification of the semicubical parabola. His procedure in this con- 
nection is typical of his general approach and indicates well the inter- 
relation of the various aspects of his work. For any point P on the 
curve with abscissa OQ — a and ordinate PQ = b, the subtangent, 
TQ = c, is known by his tangent method to be c = \a (fig. 21). If 
then an ordinate P'Q' to the tangent line is erected at a distance E 
from the ordinate PQ, the segment PP' is known in terms of a and E. 

[9a 
For the curve ky 2 = x?, this is PP' = EiJ — + 1. But the point 

m Simon appears to be overenthusiastic in saying of this work ("Geschichte und Phil- 
osophic der Differentialrechnung," p. 119): "F. hatte auch bereits in ahnlicher Weise wie 
spater Riemann den Begriff des bestimmten Integrals erfasst bei der Berechnung von 

* 

fx?dx. Hier ist Grenziibergang, hier ist Bestimmung des Werthes -, bier is volliges Be- 

wusstsein des continuitats gesetzes." 

223 Kaestner, Gesckichte der Mathematik, I, 498; III, 283. 

224 See Zeuthen, "Notes," 1895, pp. 73-76; and Pascal, (Euvres, VIII, 145, IX, 201. 
m Moritz Cantor, Vorlesungen, II, 827 ff . 



A Century of Anticipation 



163 



P', for small values of E, may be regarded as on the curve as well as 

on the tangent line, so that the length of the curve can be thought of 

as the sum of segments such as PP' . The sum of these segments, in 

9x 
turn, can be taken as the area under the parabola y 2 = -— + 1. Inas- 

4k 

much as the quadrature of this is known, the length of the curve is 
determined. 226 

It is surprising that Fermat, who used his method of maxima and 
minima for finding centers of gravity, who reduced a problem of recti- 
fication which involved tangents to a question of quadratures, who 




O T Q 

FIGURE 21 

used infinitesimals geometrically and analytically in such a wide 
variety of problems, should have missed seeing, as Pascal had also, 
the fundamental connection between the two types of questions. Be- 
cause these two men did not see this, in problems in which integrations 
by parts would now be employed they had recourse to clever geo- 
metrical transformations. Fermat, in his problems, made use of dia- 
grams which are much the same as the one of Pascal's which Leibniz 
later found so suggestive of his differential triangle, and yet he did not 
perceive their deeper significance. Had Fermat only observed more 
"• For Fermat's rectification, see (Euvres, I, 211-54; m, 181-215. 



164 A Century of Anticipation 

closely the results for the tangents and the quadratures of his parab- 
olas and hyperbolas, he might have discovered the fundamental 
theorem of the calculus and have become, what he has sometimes been 
unwarrantedly called, the "true inventor of the calculus." 227 

Fermat, of course, realized in a sense that the two types of problems 
had an inverse relationship. That he did not pursue this thought 
further may well have been due to the fact that he thought of his 
work simply as the solution of geometrical problems and not as rep- 
resenting a type of argument significant in itself. His methods of 
maxima and minima, of tangents, and of quadratures he regarded as 
constituting characteristic approaches to these questions, rather than 
a new type of analysis. Furthermore, they were apparently restricted 
in application. Fermat knew how to make use of them only in the 
case of rational expressions, whereas Newton and Leibniz, through 
their application of infinite series, recognized the universality of such 
procedures. Nevertheless, no mathematician, with the possible ex- 
ception of Barrow, so nearly anticipated the invention of the calculus 
as did Fermat. 

The influence of Fermat on his contemporaries and immediate suc- 
cessors 228 is difficult to determine. Probably his work was not so well 
known as that of Cavalieri, for the latter was widely read in his two 
famous books, whereas Fermat did not publish either his methods or 
his results. For this failure to publish his work, Fermat, it has been 
said, 229 lost the credit for the invention of the calculus; but such an 
assertion is incorrect. In the first place, it is clear that he cannot be 
thought of as its inventor. Secondly, his work was collected and printed 
posthumously as the Opera varia, in 1679, before either of the earliest 
published accounts of the calculus by Newton or Leibniz had appeared. 

In spite of the fact that Fermat did not himself publish his methods, 

227 Lagrange, Laplace, and Fourier have so called him, but Poisson has correctly pointed 
out that Fermat does not deserve such a designation, inasmuch as he failed to recognize 
the problem of quadratures as the inverse of that of tangents. The relevant statements 
of these four men may all be found in Cajori, "Who Was the First Inventor of the Calcu- 
lus?" Cf . also Marie, Histoire des sciences mathematiques, IV, 93 ff. Sloman has very unfairly 
said that "Fermat hardly deserves to be named at all" in this connection. See his Claim of 
Leibnitz to the Invention of the Differential Calculus, pp. 45-47. 

228 See Genty, L 'Influence de Fermat sur son siicle. This book is more concerned, however, 
to show the priority and independence of Fermat's results than specifically to point out 
their effects on others. 

229 Dantzig, Number, the Language of Science, pp. 131-32. 



A Century of Anticipation 165 

these became known through his correspondence with such men as 
Roberval, Pascal, and Mersenne, as well as through the publication by- 
others during his lifetime of portions of it. As a consequence, his work 
was in large part responsible for a number of transition methods which 
appeared just before the advent of the calculus. It is therefore scarcely 
correct to say, with Lagrange, 230 that "Fermat's contemporaries did 
not seize the spirit of this new type of calculus." Infinitesimal con- 
siderations such as those of Fermat constituted a large portion of the 
mathematical activity of the period. Nevertheless, there was indeed 
one great mathematician who remained somewhat cool toward the 
new views, even though in his early years he had made effective use 
of them. This was Rene Descartes, the severest critic of Fermat. 

The very first mathematical production of Descartes was an attempt, 
in 1618, to deal with the laws of falling bodies by means of infinitesimals. 
In this he made an error, in that he assumed, as Galileo had also in 
1604, 281 that the velocity was proportional to the distance rather than 
the time; but if the distance axis in his demonstration were changed to 
the time axis, his procedure would be essentially that which Oresme 
and other Scholastics had used. 232 Descartes was probably familiar 
with their works and may have derived this form of proof, as well as 
suggestions on analytic geometry, from reading Oresme. 233 

At any rate, Descartes was acquainted with ancient, medieval, and 
modern views on infinitesimals, and used them. In a second memoir of 
about the same time he wrote on fluid pressure. In this connection he 
may have known through Beekman of Stevin's work on infinitesimals. 
At all events, in considering the force drawing a body, he used such 
phrases as the "first instant of its movement," and the "first imaginable 
speed." 234 Some years later, in 1632, he answered correctly a number of 

00 See Brassine, PrScis des ceuvres math&matiques de P. Fermat, p. 4. 

ni Duhem, fitudes sur Lionard de Vinci, III, 564. 

m See Descartes, (Euvres, X, 219; cf. also X, 59, 76-77. 

*» There is a great difference of opinion on this subject. Wallner ("Entwickelungsge- 
schichtliche Momente," p. 120) sees not the least influence of Oresme on Descartes; 
Stamm ("Tractatus de continuo," p. 24) says that the problem of the latitude of forms of 
Oresme was undoubtedly the most important influence on Descartes; Wieleitner ("Ueber 
den Funktionsbegriff," p. 242) says that Descartes undoubtedly knew of Oresme's work, 
but that the essential idea of the dependence of quantities found in analytic geometry is 
missing in Oresme; Duhem (Etudes sur Lionard de Vinci, HI, 386) has claimed that Oresme 
created analytic geometry. 

«* Milhaud, Descartes savant, pp. 162-63. 



166 A Century of Anticipation 

questions, which Mersennc had sent him, on areas, volumes, and 
centers of gravity connected with the parabolas y n = px — problems 
similar to those which Fermat had solved. Descartes did not tell what 
method he had used, but it was probably a skillful application of the 
methods of Archimedes, Kepler, and Cavalieri. 235 However, after the 
publication of his famous Geometrie in 1637, Descartes' interest in 
the subject began to wane, 236 for his mathematical work was only an 
episode in the development of his philosophy. Consequently he did not 
participate effectively in the development of the infinitesimal methods 
which preoccupied the minds of most mathematicians at this time. 237 
The acrimonius quarrel with Fermat, however, sustained his interest 
in the problem of tangents and led him to considerations which, if 
pursued further, might have been more effective than the infinitesimal 
methods in leading to a clearer understanding of the basis of the cal- 
culus. 

The ancients, with the possible exception of Archimedes, had not 
developed a general definition of a tangent to a curve nor any method 
of determining it. Descartes, however, realized more fully than a 
number of his contemporaries that this constituted not only "the 
most useful and general problem that I know but even that I have 
ever desired to know in geometry. " 238 He thereupon elaborated his 
celebrated method of tangents in terms of the equality of roots. 
Descartes' method consisted in passing through two points of the 
curve a circle with its center on the x axis, and then making the points 
of intersection coincide. The center of the circle thus becomes the 
point on the x axis through which the normal to the curve passes, and 
the tangent is consequently known. This may be illustrated in some- 
what simplified form as follows: Let the tangent at the point (a, a) on 
the parabola y 2 = ax be required. The equation of the circle going 
though (a,a) and having its center on the axis is x 2 + y 2 — Ihx + 2ah 
— 2a 2 = 0, where h is undetermined. Substituting ax for y 2 , the quan- 
tity h may then be so determined that the resulting equation has equal 
roots — that is, so that the intersections of the circle with the parabola 

*" Ibid., pp. 164-68. 

at Ibid., p. 246; cf. also Marie, Histoire des sciences mathimatiques et physiques, IV, 21. 
m Milhaud (Descartes savant, pp. 162-63) denies that it was the insistence on clear ideas 
which made Descartes avoid the use of infinitesimals, 
w Descartes, (Euvres, VI, 413. 



A Century of Anticipation 167 

coincide. 239 This value, h = fa, is the abscissa of the point on the axis 
through which the normal to the parabola passes. The tangent is then 
the line through (a, a) perpendicular to this normal. 

It is to be remarked that Descartes' method is purely algebraic, no 
concepts of limits or infinitesimals being manifestly involved. However, 
any attempt to interpret geometrically the significance of the case in 
which the roots are equal, to explain what is meant by speaking of co- 
incident points, or to define the tangent to a curve, would necessarily 
lead to these conceptions. If Descartes in his geometry had thought in 
terms of continuous variables rather than of a correspondence between 
symbols which represented lines in a geometrical diagram, 240 he might 
have been led to interpret his tangent method in terms of limits, and 
so have given a different direction to the anticipations of the calculus. 
His algebra, however, was still grounded in the geometry of lines, and 
the idea of a continuously varying quantity was not really established 
in analysis until the time of Euler. 241 

In criticising Fermat's method of tangents, Descartes attempted to 
correct the method by interpreting it in terms of equal roots and co- 
incident points, a procedure which was practically equivalent to de- 
fining the tangent as the limit of a secant. 242 Descartes did not express 
himself in this manner, however, inasmuch as the concept of a limit 
was far from clear at this time. Fermat, who was thinking of infinitesi- 
mals, could not see that his method had anything in common with the 
algebraic (limit) method of Descartes and so precipitated a quarrel as 
to priority, one of the many which the seventeenth century produced 
as the result of the confusion of thought as to the basis of infinitesimal 
methods. Descartes preferred his method of tangents to that of Fermat 
because of the apparent freedom from the concept of infinitesimals, al- 
though its application was frequently much more tedious and was 
limited to algebraic curves. 

In finding the tangent to a nonalgebraic or "mechanical" curve, such 
as the cycloid, Descartes in 1638 made use of the concept of an in- 
stantaneous center of rotation. This is, of course, also directly con- 
nected with the use of limits and infinitesimals, but is expressible 

m Ibid., VI, 413-24; cf. also Voss, "Calcul differentiel," p. 244. 

"• Descartes, (Euvres, VI, 369; cf. also VI, 411-12. 

^ Fine, Number-System, p. 121. 

"" See Duhamel, "M6moire," pp. 298-308; Milhaud, Descartes savant, pp. 159-62. 



168 A Century of Anticipation 

without the use of such terminology, through the circumlocution af- 
forded by the notion of instantaneous velocity. Supposedly, this notion 
is intuitively clear, but at the time it was not rigorously defined. It 
had been made acceptable, moreover, by the work of Galileo, 243 and 
Roberval and Torricelli were employing it in geometry at practically 
the same time as Descartes. Descartes' reasoning was as follows: If a 
polygon is rolled along a straight line, any vertex will describe a series 
of circular arcs, the centers of which are the points on the line which 
the vertices of the polygon touch: i. e., in rolling the polygon along the 
line, we rotate the polygon about each of these points in turn. The 
cycloid, now, is the curve generated by a point on a circle: i. e., by a 
vertex of a polygon with an infinite number of sides, as it is rolled 
along a line. The cycloid is therefore made up of an infinite number 
of circular arcs, and the tangent at any point, P is therefore perpen- 
dicular to the line joining P to the point Q in which the generating 
circle touches the base line. Inasmuch as Q can easily be determined, 
the tangent at P can be drawn. 244 

In Descartes' work one sees an avoidance of the idea of an infinitely 
small quantity in mathematics and the use, instead, of algebraic and 
mechanical conceptions. Whereas Fermat saw only the practical ad- 
vantages of the infinitesimal methods, Descartes saw better the risks 
they entailed. Descartes' evasion of them was of course justified by the 
lack of a clear theoretical basis for infinitesimal reasoning, but it was 
opposed to the mathematical trend of the time. We have seen that in 
the years following his Discours de la methode, there appeared in print 
an unprecedentedly large number of works devoted to infinitesimal 
methods. In most of them the work was largely based on synthetic 
geometry, although Roberval and Pascal showed an arithmetizing 
tendency in their quadratures. In France only Fermat made effective 
use, in the anticipations of the calculus, of the new analytic methods 
which he and Descartes were developing. In England, however, the 
mathematician and theologian John Wallis applied analytic geometry 
to the problem of quadratures with comparable success. 

John Wallis had become familiar with analytical methods largely 

241 De Giuli ("Galileo e Descartes") asserts that Descartes owed to Galileo much of his 
philosophic method. 
M Walker, A Study of the Traits des Indivisibles of Roberval, pp. 137-39. 



A Century of Anticipation 169 

through Harriot. In his work on conic sections, Wallis followed Viete, 
Descartes, Fermat, and Harriot in the application of literal algebra 
to the problems of geometry. Wallis, however, went far beyond these 
men in that he sought to free arithmetic completely from geometric 
representation, a goal which he thought would be easily reached. 246 He 
first showed how all the theorems of Euclid V could be derived arith- 
metically without difficulty, and then in algebra broke away from the 
idea, derived from geometry, that the terms of an equation must be 
homogeneous. Luckily, Wallis did not worry overmuch about mathe- 
matical rigor: we know now how difficult the arithmetizing of mathe- 
matics was to be. 

Instead of observing the caution which classical ideas of rigor ex- 
acted, Wallis was influenced by prevailing thought to make free use of 
analogy and incomplete induction in his work, as well as of the con- 
cepts of infinity and infinitesimals, which had not yet been rigorously 
established. We have seen these tendencies developing in the work of 
Cavalieri and Fermat and, in continuing this tradition, Wallis came 
nearer to the limit concept than did any other of Newton's predeces- 
sors. It is clear that this notion is implicit in the work of most of his 
French and Italian contemporaries, but it was not expressed by them. 
Instead, the concept of the infinitesimal was employed. With Wallis' 
arithmetical point of view, however, one is brought again into more 
direct touch with the limit idea which the mathematicians of the Low 
Countries — Stevin, Gregory of St. Vincent, and Tacquet — had sought 
to formulate. 

To what extent Wallis was influenced by the ideas of these other 
men is difficult to determine. He wrote the most complete treatise on 
statics since the time of Stevin, 246 and may well have been familiar 
with the arithmetical limit methods of the latter, either directly or 
through the somewhat similar work of Roberval. 247 Wallis admitted 
that upon the advice of Wren he read, about 1652, part of the Opus 
geometricum of Gregory of St. Vincent, but he added that in this he 
did not run across any propositions new to him. 248 On the other hand, 
Wallis and Tacquet were probably independent of each other in their 

245 Prag, "John Wallis." *• Duhem, Les Origines de la statique, II, 211. 

247 Walker, A Study of the TraiU des Indivisibles of Roberval, p. 77; cf. also p. 165. 

248 Wallis, Opera mathematico, Vol. II, Arithmetica infinitorum, Preface. 



170 A Century of Anticipation 

work, inasmuch as their books on this subject appeared almost simul- 
taneously — in 1655 and 1656 respectively. Oddly enough, moreover, 
although Wallis displays the arithmetizing and limit tendencies of 
Tacquet, the chief inspiration for his work came rather from reading, 
in 1650, the geometrical method of indivisibles of Cavalieri as ex- 
pounded by Torricelli. This he professed in the prefaces of two of his 
books: De sectionibus conicis tractatus, and Arithmetica infinitorum sive 
nova methodus inquirendi in curvilineorum quadraturam. 249 Cavalieri's 
work, however, had been almost purely geometrical, whereas Wallis 
proceeded largely arithmetically, and in the end abstracted from the 
geometry of indivisibles the arithmetic notion of a limit. "Following 
Oughtred, Descartes, and Harriot," he applied to his demonstrations 
the symbolism of arithmetic in order to give to them "at the same time 
the maximum brevity and perspicuity." The use of arithmetic calcu- 
lation he held to be simpler and not less "legitimate or scientific" than 
that by lines. 250 

The manner in which Wallis made the transition from the geometry 
of lines to the arithmetic of numbers is brought out clearly in his proof 
that the area of a triangle is the product of the base by half the alti- 
tude. 261 He assumed at the outset, as had Cavalieri, that a plane figure 
may be regarded as made up of an infinite number of parallel lines — 
or rather, as he preferred, of an infinite number of parallelograms, the 

altitudes of which are equal, that of any one of them being — , or an 

infinitely small aliquot part of the altitude of the figure. 252 Here we 
have not only the first appearance of the symbol o° for infinity, 253 but 
also the earliest use of the Scholastic categorematic infinity in the 
field of arithmetic. Furthermore, the treatment by Wallis of the in- 
finitely small is far more daring and decisive than that of Fermat. 
Whereas the latter had not expressly called his symbol E an infinitesi- 
mal, Wallis for his part said that _L represented an infinitely small 

oo 

quantity, or non-quanta. A parallelogram whose altitude is infinitely 

*• Opera motketnatico, Vol. II. 

260 Opera mathematica, Vol. II, De sectionibus conicis, "Dedicatio," and also p. 3. 
851 Ibid., pp. 4-9. A good summary of the work of Wallis along these lines is to be found 
in Sloman, The Claim of Leibnitz to the Invention of the Differential Calculus, pp. 8 ff . 
** Opera mathematica, Vol. II, De sectionibus conicis, p. 4. 
m "Esto enim oo nota numeri infiniti." Ibid. 



A Century of Anticipation 171 

small or zero is therefore "scarcely anything but a line," except that 
this line is supposed "extensible, or to have such a small thickness that 
by an infinite multiplication a certain altitude or width can be ac- 
quired." 264 

Returning to the proposition on the area of the triangle, Wallis sup- 
posed this to be divided into an infinite number of lines, or infinitesimal 
parallelograms, parallel to the base. The areas of these, taken from the 
vertex to the base, form an arithmetic progression beginning with zero. 
Moreover, there is a well-known rule that the sum of all the terms in 
such a progression is the product of the last term by half the number 
of terms. Since "there is no cause for discrimination between finite and 
infinite numbers," it can be applied to the areas in the triangle. If the 
altitude and base of the triangle are taken as A and B respectively, the 
area of the last parallelogram in the progression will then be — A3. 

1 °° 

The area of the whole triangle is therefore J— A 3. 9 , or \A3. m He 

00 * 

then applied a similar type of argument to numerous quadratures and 
cubatures involving cylinders, cones, and conic sections. 

Wallis realized that his procedure was highly unorthodox, but he 
said that it could be verified by "that very well-known apagogic 
method" of in-and-circumscribed figures. To give this Wallis felt 
would be superfluous, because "the frequent iteration would create 
nausea in the reader." Furthermore, he said that anyone versed in 
mathematics could supply such proof, since it occurred frequently 
among the ancients and the moderns. 256 Modern mathematics has 
found it necessary to modify greatly the view of the infinite which 
Wallis held and to banish entirely his infinitely small magnitudes. 
Nevertheless, the development of the calculus is the result of efforts, 
such as his, to substitute for the prolixity of the method of exhaustion 
a direct arithmetical analysis. 

The procedure of Wallis in De sectionibus conicis was based largely 
upon crude manipulations of his symbol o° . In the Arithmetica in- 
finitorum, however, he pursued similar investigations from a somewhat 
different point of view — one resembling more closely the arithmetic 

*"Ibid. 

156 Ibid., pp. 8-9; cf. also Opera maihematico, Vol. II, Arithmetica infinitorutn, p. 2. 

*• Opera mathematica, Vol. II, De sectionibus conicis, p. 6. 



172 A Century of Anticipation 

methods of Stevin and Roberval and the limit concept. In this con- 

nection he demonstrated the equivalent of the theorem : JqX dx = , 

« + 1 

apparently unaware that this proposition had appeared in numerous 

forms during the preceding twenty years. Wallis reached this conclusion 

by observing first the equalities 

+ 1 = ,. + 1 + 2 = , + 1+2 + 3 = le 

1 + 1 2 '2+2 + 2 2 '3 + 3 + 3 + 3 " 

In these the ratio is \ for any finite number of terms, from which Wallis 
concluded that this will be the ratio likewise for an infinite number. 
Through this, Wallis arrived at an alternative form of demonstration 
for the theorem above on the area of a triangle. 257 

In proceeding further, Wallis noted that in the equalities 
+ 1 1 . + 1+4 _ 1 , ,, + 1+4 + 9 , , . 

nri-' + *>4 + 4 + 4" 5+ ™'9 + 9 + 9 + 9~ 5 + Tlr '---' 
the greater the number of terms, the more closely does the ratio ap- 
proximate to |, so that "at length it differs from it by less than any 
assignable magnitude." If this is continued to infinity, the difference 
"will be about to vanish completely." Consequently, the ratio for an 
infinite number of terms is i. 268 

Wallis then proceded to observe, in a similar manner, that the 
analogous ratios for the third, fourth, fifth, and higher powers of the 
integers are respectively \, i, i, and so on. He then affirmed the validity 
of the rule for all powers, rational or irrational (except, of course, 
— I). 269 This extension of the work of his French and Italian prede- 
cessors was made upon the basis of what Wallis spoke of as interpola- 
tion and induction. By the former, he seems to have had in mind a 
principle of continuity, 260 or of permanence of form, by which the rule 
could be asserted to hold for values intermediate between those for 
which it was known to be valid. By the latter, Wallis meant not mathe- 
matical or complete induction but induction in the scientific sense, 

267 Opera mathematica, Vol. II, Arithmetica infinitorum, pp. 1-3; cf. also p. 157. 

™Ibid., pp. 15-16; cf. also p. 158. 

•*• Ibid., pp. 31-53. Wolf 04 History of Science, Philosophy, and Technology in the Six- 
teenth and Seventeenth Centuries, p. 209) has incorrectly stated in this connection that the 
predecessors of Wallis confined themselves to positive integral powers. Furthermore, he 
neglects to state that Wallis extended the rule to irrational powers also, 

»° Nunn, "The Arithmetic of Infinities." 



A Century of Anticipation 173 

analogy similar to that by which we have seen him conclude properties 
of the infinite from those of the finite. In this respect his work is a 
good indication of the looseness of thought at the time. More signifi- 
cantly, his extension of the rule to irrational powers indicates a tend- 
ency to break away from the persistent idea, derived from Pythag- 
orean geometry, that irrational magnitudes are not numbers in the 
strict sense of the word. This was in line with his declaration of the 
independence of arithmetic from geometry, a freedom which was neces- 
sary for the later elaboration of the limit concept which he was here 
adumbrating. 

The proposition above, on the ratios of the powers of integers, Wallis 
then applied to problems on quadratures and cubatures. In this re- 
spect he may be said to have determined the areas and volumes as 
limits of infinite sequences, in much the same way as Fermat had found 
them by means of infinite geometric progressions. In fact, the basis for 
the concept of the definite integral may be considered fairly well estab- 
lished in the work of Fermat and Wallis, although it was to become con- 
fused later by the introduction of the conceptions of fluxions and differ- 
entials. However, that neither of these men realized fully the significance 
of this concept is seen in their lack of clarity. We have seen that Fermat 
did not fully explain the nature of his symbol E. Wallis confused his 
work with the infinitesimal, identifying infinitely small rectangles with 

lines, and writing J_ = — ideas which were to lead to the conception, 

oo 

found in Leibniz, of the integral as a sort of totality, rather than as the 

limit of a sum. 

Wallis was interested in another question which characteristically 

concerned mathematicians of the time — that of the angle of contact 

(horn angle) formed by two curves with a common tangent, a figure 

which Euclid, Jordanus Nemorarius, Cardan, and many others had 

considered. Discussion on this point may have been instrumental in 

maintaining the concept of the infinitely small as a valid notion, for 

it lent plausibility to the idea of an ultimate indivisible, smaller than 

any assignable magnitude and yet seemingly not the same as absolute 

zero. 261 The postulate of Archimedes of course excluded such angles 

261 For a full discussion of the history of this subject, see Vivanti, 77 concetto, or the French 
summary of this book given in Bibliothica Mathematica, N. S., VIII (1894) as "Note sur 
l'histoire de Pinriniment petit." 



174 A Century of Anticipation 

as magnitudes, just as it had excluded other infinitesimals; but mathe- 
maticians of the seventeenth century regarded them as interesting il- 
lustrations of their concepts, and discussed the question as to whether 
they were zero or not. Galileo, Wallis, 262 and others asserted that such 
angles were absolutely zero, whereas Hobbes, Leibniz, and Newton, 
for example, held that they were in some way different from zero. On 
this point argument was possible, of course, only for two reasons: the 
general lack of critical definitions during this period (a rigorous defi- 
nition of the tangent to a curve, although implied by the work of 
Descartes, Fermat, and others, having not been given at the time), and 
the failure to distinguish clearly between the geometrical figure and 
its arithmetical measurement. Both of these shortcomings were to be 
significant later in exposing the calculus of Newton and Leibniz to 
severe criticism. 

After a century of doubt, clear definitions were formulated and the 
calculus was established upon arithmetical rather than geometrical 
conceptions. The work of Wallis was an attempt to bring about such 
an arithmetization, and in this respect it won the support of his con- 
temporary, James Gregory. The latter, in his Vera circuli et hyperbolae 
quadratura of 1667, viewed the passage to the limit as an independent 
arithmetical operation, suitable as a means of defining new numbers 
not belonging to the ordinary irrationals. 263 In connection with this 
work, he constructed in and circumscribed polygons to the circle and 
hyperbola and showed that, by doubling the number of sides of these, 
converging series were obtained in which the difference became smaller 
and smaller. These series consequently had a limit, which, "if one may 
speak in this manner," could be considered the last polygon in each 
series. This consequently would give the area of the curvilinear figure. 264 
The areas of the circle and the hyperbola Gregory gave to as many as 
twenty-six figures, although the limit he recognized as in general in- 
commensurable. 266 

This work on the limit of converging infinite series represented a 
generalization of the earlier propositions of Gregory of St. Vincent and 
Tacquet on geometric progressions, with which James Gregory was 

M * Opera mathematica, Vol. II, De angulo contactus et semicirculi. 
*«* Wallner, "Uber die Entstehung des Grenzbegriffes," p. 258; cf. also Georg Heinrich, 
"James Gregorys 'Vera circuli et hyperbolae quadratura.' " 
*•* Vera circuli et hyperbolae quadratura, pp. 15-16. 2M Ibid., pp. 48, 25. 



A Century of Anticipation 175 

acquainted. 266 He may also have been familiar with the somewhat 
similar arithmetic work of Roberval, for in another connection he 
used geometric transformations resembling the so-called Robervallian 
lines. 267 However, whereas Roberval and Wallis had been led to their 
arithmetization through the method of indivisibles, Gregory preferred 
to employ in quadratures the indirect method of the ancients, showing 
that the difference can be made less than any given quantity. 268 
Nevertheless, in connection with this work he adopted the newer ana- 
lytic methods of Descartes. In this respect also he followed the method 
of Fermat for determining tangents to curves. For example, the tangent 
to y* = x 2 (a + x) he found at the point for which x = b as follows: 269 
Choose a second point, with abscissa less than a; by a vanishing small 
amount, o, 270 and assume, "if we may do so," that the corresponding 
ordinate may be taken indifferently as that of the curve or of the tan- 
gent. Then set up the suitable proportion, divide by o, and reject the 
terms in which o or a power of it remain. The subtangent is in this way 

3b 2 + 3ab 

found as z = . 

3b + la 

Although Gregory did not refer to Fermat in this connection, it is 
obvious that the methods of the two men are identical with the ex- 
ception that the E of Fermat has been changed to o, a change of no- 
tation which was to be adopted a year or two later by Newton, perhaps 
under the influence of Gregory's work. The type of arithmetical and 
analytical work of Fermat, Wallis, and Gregory represented the tend- 
ency which was to lead to the calculus, but it met with almost im- 
mediate opposition, for the spirit of the age was directed toward the 
solution of problems through geometrical considerations. Such an 
arithmetization of mathematics was opposed with particular vigor by 
two Englishmen, the philosopher Thomas Hobbes and the mathe- 
matician and theologian Isaac Barrow. 271 Hobbes objected strenuously 
to "the whole herd of them who apply their algebra to geometry." 272 

»• Ibid., p. 20; cf. also p. 123. M7 Galloys, "Reponse a l'Scrit de M. David Gregorie." 

468 See Geometriae pars universalis, pp. 27-29, 74 ff. 

»• Ibid., pp. 20-22. » "Nihil seu serum o." Ibid. 

m See Cajori, "Controversies on Mathematics between Wallis, Hobbes*, and Barrow." 
A summary of this article is given in Bulletin, American Mathematical Society, XXXV 
(1929), 13. 

272 Weinrich, Ober die Bedeutung des Hobbes fur das naturwissenschafiliche und mathe- 
matiscke Denken, p. 91. 



176 A Century of Anticipation 

He maintained that they mistook the study of symbols for that of 
geometry, and characterized the Arithmetica infinitorum as a "scurvy 
book." 273 He referred to the arithmetization represented by Wallis as 
absurd and as "a scab of symbols." 274 

This attitude toward algebra and analytic geometry was probably 
the result not only of the general predilection in the seventeenth 
century for geometrical rather than arithmetical methods, but also of 
Hobbes' exaggerated view of mathematics as an idealization of sensory 
perception, rather than as a branch of abstract formal logic. Greek 
thought had accepted mathematics as derived from the experience of 
the senses by the abstraction from concrete objects of irrelevant 
properties. Hobbes, however, was unwilling to regard lines as deprived 
of all breadth, or surfaces of all thickness. 276 Consequently, the in- 
finitely small was for him merely the smallest possible line, plane, or 
solid — a view of infinitesimals held by the school of mathematical 
atomists in antiquity and not unlike that of Cavalieri. 

Hobbes' view of number was comparable to his attitude toward the 
geometrical elements. He adopted the Pythagorean notion of number 
as a collection of units, and he interpreted ratios only in terms of 
geometrical considerations. 276 This attitude would not only operate 
against the free use of arithmetic processes which Wallis carried out on 
the basis of analogy or induction, but would, in fact, when combined 
with Hobbes' naive view of geometrical magnitude, make the dis- 
tinction between rational and irrational number, and the consequent 
introduction of the limit concept, logically superfluous. Hobbes' ideas 
in this direction were, as a result, of little consequence in the develop- 
ment of the concepts of the calculus. The.e is, however, another aspect 
of his thought which may have been more significant. 

We have mentioned that Aristotle thought of motion as the fulfill- 
ment of the potential, thus centering attention not on the mathematical 
aspect of motion, but rather on the metaphysical, attributing to a body 
in motion a striving toward a goal. It could not of itself, however, reach 
its goal, since, in order that it do so, the constant application of a force 
was necessary. In the fourteenth century this Peripatetic theory was 
questioned, because it could not "save the phenomena." In its place, 

« English Works, VII, 283. K4 Cf. ibid., VII, 187, 361 ff. 

» Ibid., VII, 67, 200 ff., 438. m Cf. Opera omnia, IV, 27, 36. 



A Century of Anticipation 177 

Jean Buridan had substituted the doctrine of impetus (or inertia), or 
the tendency of a body to remain in motion. 277 

This Scholastic view of impetus gave to motion a so-called intensive 
characteristic, for it centered attention upon the act of moving, rather 
than on change of position or extension. Such a shift of emphasis made 
acceptable the notion of motion at a point, an idea which Aristotle 
had specifically rejected. 278 This was followed immediately by the 
quantitative treatment of instantaneous velocity found in Calculator, 
Hentisbery, and Oresme. The idea of impetus was familiar to Nicholas 
of Cusa, Leonardo da Vinci, and others of the fifteenth and sixteenth 
centuries, although it was often associated with, and obfuscated by, 
accretions of Neoplatonic mysticism and of vitalistic and teleological 
thought, such as are to be found in the works of Nicholas of Cusa, 
Paracelsus, and Kepler. 279 In 1638 the idea of impetus or inertia cul- 
minated in the famous clarification of the laws of dynamics given by 
Galileo; and at the same time the concept of instantaneous velocity 
was likewise successfully applied in geometry by Torricelli, Roberval, 
and Descartes. 

It must be kept in mind, of course, that no definition of instanta- 
neous velocity had at the time been given, nor could it have been 
framed before the development of the differential calculus. However, 
the concept of inertia was to make the idea of motion at a point in- 
tuitively and scientifically acceptable — as well as philosophically in- 
teresting and geometrically useful — until such time as it could be made 
mathematically rigorous. 

Philosophers have generally displayed an interest in the calculus 
because of its associations with problems of motion and variation which 
present many intriguing metaphysical aspects. Hobbes was especially 
concerned, inasmuch as he wished to make motion basic in his philo- 
sophical scheme. His views were mathematically too naive to allow 
of his adding to the discoveries being made in the new analysis, al- 
though he may have had a somewhat broader influence on the later 

m Duhem, Etudes sur Leonard de Vinci, III, vii-viii. m Physica VI. 234a. 

2,9 The great German astronomer, for example, in his Mysterium cosmographicum and 
his Astronomia nova, ascribed a vital faculty to the sun and the planets. In the 1621 
edition of the former work, however, he said that one can substitute the word force for 
soul. See Opera omnia, I, 174; II, 270; III, 176, 178-79, 313. See also Lasswitz, Geschichte 
der Atomistik, II. 9-12; Duhem, Etudes sur Leonard de Vinci, II, 199-223. 



178 A Century of Anticipation 

thought and interpretation of the calculus. Hobbes was much im- 
pressed with the success of Galileo's conception of the laws of motion, 
in terms of inertia and changing velocity. Realizing that physics owes 
its development to mathematical representation, 280 he wished to de- 
scribe and explain these ideas in geometrical and metaphysical terms. 
To this end, he introduced the concept of the conatus, 2 * 1 a beginning of 
motion, analogous to the concept of the point as the beginning of 
geometrical extension. Hobbes thus tried to emphasize, as had the 
men of the fourteenth century, the idea of motion at a point, rather 
than that of change of position. Not realizing, as Aristotle had dis- 
cerned nineteen hundred years before, that instantaneous motion is 
an intellectual — not an empirical — concept, he attempted to formulate 
a definition in terms of his ingenuous nominalism, speaking of it as the 
motion in an infinitely small interval — an interval less than any given 
interval — that is, through a point. 282 

Hobbes was not so fortunate, as Newton and Leibniz were later, in 
formulating mathematically his concept of conatus. He did not under- 
stand the relation between number and spatial quantity, nor did he 
appreciate that instantaneous velocity is a purely numerical notion. 
His views, however, were perhaps significant in their influence upon 
the inventors of the calculus. The excessive nominalism of Hobbes was 
to lead mathematicians away from a purely abstract view of the con- 
cepts of mathematics, such as Wallis had displayed, and to induce them 
to seek, during more than a century, for an intuitively, rather than a 
logically, satisfactory basis for the calculus. It was largely on this ac- 
count that both Newton and Leibniz sought to explain the new an- 
alysis in terms of the percipient notion of the generation of magnitude, 
rather than in terms of the logical conception of number only. 283 This 
idea of generation is more immediately apparent in the method of 
fluxions of the English empiricist, Newton ; but the German philoso- 
pher, Leibniz, justified his differential method also in terms of the 
analogous idea of continuity. Whereas Newton used the physical idea 
of the "moment" of growing magnitudes, there grew up in Germany 
a more metaphysical form of this in the notion of intensive magnitude 
as opposed to extensive quantity. Upon this idea of a "tendency" or a 

280 Opera omnia, II, 137. » Opera omnia, I, 177. «* Ibid. 

283 Cf. Vivanti, // concetto d'infinitesimo, pp. 31-32. 



A Century of Anticipation 179 

"becoming," the mathematical infinitesimal throve, with the result 
that philosophers have been reluctant to abandon it, even though 
modern mathematics has shown that the basis of the calculus is to be 
found in the derivative rather than in the differential. 

There is, furthermore, an inclination on the part of philosophers (and 
occasionally of mathematicians) to regard Hobbes' conatus, or the de- 
rived concept of the intensive in motion, as the answer to Zeno's 
paradoxes, inasmuch as even when the time interval has disappeared, 
the tendency toward motion remains. 284 This attitude adds nothing to 
the explanation of the paradoxes, for it fails to recognize that the con- 
ception of motion at a point, which is the crux of the situation, is not 
a scientific notion but a mathematical abstraction. As such, the logical 
difficulties involved have all been cleared up by the calculus and the 
mathematical continuum. Of course, there is nothing lost (or gained) 
by calling the derivative (which is, after all, only a number) the in- 
tensity at the point, for this is only a change of name; but the answer 
to the paradoxes must remain that given by mathematics, rather than 
any made in an attempt to satisfy intuition. 

Wallis' arithmetization (as well as Descartes' analytic geometry) 
was criticized also by the contemporary mathematician, Isaac Barrow. 
The latter wished to return to the Euclidean view, and maintained that 
mathematical number has no real existence proper to itself and inde- 
pendent of continuous geometrical quantity. Numbers like ^3, he 
felt, cannot, even in thought itself, be abstracted from all magnitude. 
Such "surd" numbers are "inexplicable," and, "having no merit of 
their own, they are wont to be banished from arithmetic to another 
science (which yet is no science), viz., algebra." Barrow held that 
arithmetic is to be included in geometry, but that algebra is to be in- 
cluded in logic rather than mathematics. 285 

This view would, of course, lead directly away from the limit con- 
cept which requires, for its effective use, as well as its logical defini- 
tion, a conception of number not based on the geometrical interpre- 
tation of continuous magnitude. The fact that Barrow advocated a 
return to the classic conception of number and geometry may have in- 

184 See Lasswitz, Geschichte der Atomistik, II, 30; Vivanti, II concetto d'infinilesimo, pp. 
31-32. 
m See Barrow, Mathematical Works, pp. 39, 45-46, 51-53, 56, 59. 



180 A Century of Anticipation 

fluenced his student, Newton, to seek to establish the calculus on the 
idea of continuous variation found in motion and geometry, and to 
avoid as far as possible the arithmetical notion of limit. Barrow's dis- 
trust of algebraic methods may well have been responsible also for the 
fact that he did not develop his geometrical discoveries into an effective 
analytical tool, but in this respect he was not followed by Newton. 

Although Barrow did not accept readily the algebra which Italian 
mathematics had developed, he was impressed by the possibilities 
offered by the conception of motion at a point, which had been used 
to such advantage in geometry by Torricelli and others. Time he re- 
garded as a mathematical quantity measurable by, although not de- 
pendent on, motion, 286 and upon the suggestion of sensory evidence he 
thought of it as a continuous magnitude, "passing with a steady flow." 
This brought Barrow to the problem of the nature of the continuum 
and the definition of instantaneous velocity. In his treatment of these 
there is a complete lack of the limit concept and, instead, an attempted 
blending of atomistic and kinematic views which may have been in 
part the result of the influence of the Cambridge Platonists. "To every 
instant of time, or indefinitely small particle of time, (I say instant or 
indefinite particle, for it makes no difference whether we suppose a 
line to be composed of points or of indefinitely small linelets; and so in 
the same manner, whether we suppose time to be made up of instants 
or indefinitely minute timelets) ; to every instant of time, I say, there 
corresponds some degree of velocity, which the moving body is con- 
sidered to possess at the instant." 287 This passage shows clearly that 
Barrow's views are essentially infinitesimal and not much clearer than 
those of Plato, Oresme, Galileo, or Hobbes. In fact, in his argument 
that the area under a velocity-time curve represents distance, his 
thought closely parallels that implied by Oresme and expressed by 
Galileo. "If through all points of a line representing time are drawn. . . 
parallel lines, the plane surface that results as the aggregate of the 
parallel straight lines, when each represents the degree of velocity 
corresponding to the point through which it is drawn, exactly corre- 
sponds to the aggregate of the degrees of velocity, and thus most con- 
veniently can be adapted to represent the space traversed also." 288 

"• Gunn, The Problem of Time, p. 57. 

» Barrow, Geometrical Lectures, p. 38. ** Geometrical Lectures, p. 39. 



A Century of Anticipation 181 

Barrow admitted that it may be contended rightly that very narrow 
rectangles should be substituted for the lines, but maintained that "it 
comes to the same thing whichever way you take it." 289 

In pointing out that "time has many analogies with a line," Barrow 
again suggested atomistic conceptions, saying that these magnitudes 
could be considered as constituted either from the continuous flow of 
one instant or point, or as an aggregation of instants or points. 290 Of 
all the ways in which a continuous magnitude may be generated, 
Barrow believed, with Cavalieri, that the one in which it is regarded 
as composed of indivisibles, "in most cases is perhaps the most expe- 
ditious of all, and not the least certain and infallible of the whole 
set." 291 We have seen that Tacquet had attacked the method of in- 
divisibles, substituting for it a sort of limit of infinitely small quan- 
tities, and that Pascal followed him in this. Barrow, on the other hand, 
defended Cavalieri 's method against the valid criticisms of Tacquet. 292 

In spite of the lack of clarity and precision in his views on the con- 
tinuum, the geometrical results of Barrow represent a remarkably 
close approach to those of the calculus. They include not only nu- 
merous theorems on quadratures and tangents, but also perhaps the 
clearest recognition up to that time of the significance of the relation- 
ship between these two types of problems. 293 All of his propositions, 
however, are cast in geometrical forms which involve intricate and 
unnatural constructions, instead of in the analytical symbolism of 
Descartes, Fermat, and Wallis. If they were recast in terms of the 
calculus, they would be equivalent to many of the standard rules and 
theorems on differentiation and integration, including the fundamental 
theorem of the calculus. 294 Any such attempt to interpret them in terms 
of the present analytical notations, however, would be misleading. It 
would suggest, by implication, the possession on the part of Barrow 
of concepts equivalent to those of the derivative and the integral. 
It would furthermore give to his work an analytic character which 
the original was far from exhibiting. 

With respect to the concepts involved in Barrow's geometry, we 

have seen that his views indicate a return to the vague indivisible of 

*»Ibid. «° Ibid., p. 37. ™ Ibid., p. 43. «• Ibid., pp. 44-46. 

"• See Child, "Barrow, Newton and Leibniz, in Their Relation to the Discovery of the 
Calculus." 
** See Geometrical Lectures, pp. 30-32. 



182 A Century of Anticipation 

Cavalieri, rather than progress toward the development of the limit 
concept fostered by Wallis. As to the form of his work, it is clear that 
Barrow himself failed to appreciate it as a new type of analysis, suffi- 
ciently significant in itself to be developed into an algorithmic pro- 
cedure. He seems to have realized that he was indicating a new method 
for finding tangents and areas, but by presenting it in synthetic form 
he gave it the appearance of being an expansion of the classical ge- 
ometry of the ancients. "These matters seem not only to be somewhat 
difficult compared with other parts of Geometry, but also they have 
not been as yet wholly taken up and exhaustively treated (as the other 
parts have)." 295 

There is one point in Barrow's Geometrical Lectures, nevertheless, at 
which there is an indication that he may have used an analytic method 
to arrive at his results, recasting these later into the synthetic form in 
which he presented them. Also, in this connection, he constructed a 
diagram which was to become significant in the calculus of Leibniz as 
the familiar differential triangle. However, similar figures had ap- 
peared in the geometry of Torricelli, Roberval, Pascal, and Fermat. 
At the close of Lecture X, Barrow said: 

We have now finished in some fashion the first part, as we declared, of 
our subject. Supplementary to this we add, in the form of appendices, a 
method for finding tangents by calculation frequently used by us. Although 
I hardly know, after so many well-known and well-worn methods of the 
kind above, whether there is any advantage in doing so. Yet I do so on the 
advice of a friend [later shown to be Newton 296 ]: and all the more willingly, 
because it seems to be more profitable and general than those which I have 
discussed. 297 

If Barrow had believed that he was inventing a new subject, it seems 
likely that it is this method which he would have put forward, instead 
of his classical proofs. 

Let AP % PM be two straight lines given in position, of which PM cuts 
a given curve in M , and let MT be supposed to touch the curve at M , and 
to cut the straight line at T [fig. 22]. 

In order to find the quantity of the straight line PT,I set off an indefi- 
nitely small arc, MN, of the curve; then I draw NQ, NR parallel to MP, 
AP; I call MP = m, PT = t, MR = a, NR = e, and other straight lines, 
determined by the special nature of the curve, useful for the matter in 

"• Geometrical Lectures, p. 66. 

"• See More, Isaac Newton, p. 185, n. m Geometrical Lectures, p. 119. 



A Century of Anticipation 



183 



hand, I also designate by name; also I compare MR, NR (and through 
them, MP, PT) with one another by means of an equation obtained by 
calculation; meantime observing the following rules. 

Rule 1. In the calculation, I omit all terms containing a power of a or e, 
or products of these (for these terms have no value). 

Rule 2. After the equation has been formed, I reject all terms consisting 
of letters denoting known or determined quantities, or terms which do 
not contain a or e (for these terms, brought over to one side of the equation, 
will always be equal to zero). 

Rule 3. 1 substitute m (or MP)' for a, and t (or PT) for e. Hence at length 
the quantity of PT is found. 298 

From this passage one observes that the "method for finding tan- 
gents by calculation" used by Barrow resembles very closely the pro- 




T Q 

FIGURE 22 

cedure now employed in the differential calculus, the letters a and e 
being equivalent to the customary symbols Ay and Ax. This is an 
elaboration of Fermat's method, in which, however, only one infinitely 
small quantity, E, had been used. Barrow's method constitutes an 
improvement upon that of Fermat in that it makes more convenient 
the application of the method to implicit functions. Barrow apparently 
did not know directly of Fermat's work, for he nowhere mentioned 
his name. Nevertheless, the men to whom he referred as the sources 
of his ideas include Descartes, Huygens, Galileo, Cavalieri, Gregory of 
St. Vincent, James Gregory, and Wallis, 299 and it is possible that 
Fermat's method became known to Barrow through them. Huygens 
and James Gregory, in particular, made frequent use of Fermat's 
«• Geometrical Lectures, p. 120. "• See Geometrical Lectures, p. 13. 



-1 



184 A Century of Anticipation 

characteristic procedure; and Newton, at any rate, recognized that 
Barrow's rule was but an improvement upon this method of drawing 
tangents. 300 

Although the tangent method of Barrow resembles the process of 
differentiation even more closely than does that of Fermat, one is not 
at liberty to impute to him the concepts implied by our symbols Ay 
and Ax. He was quite evidently thinking in terms of geometrical 
problems and infinitesimals, rather than of functions and symbols for 
continuous variables. His statement that a and e "have no value" is 
equivalent in this respect to Fermat 's neglect, at the end of the cal- 
culation, of all terms involving E. Neither Barrow nor Fermat justi- 
fied the neglect of these terms, for neither one had a clear conception 
of a limit. Fermat, perhaps, came nearer to such an idea in thinking 
of his pseudo-equality as a rigorous equality only when E was zero. 
Barrow did not tell why the higher powers are to be neglected. The 
first and third rules given by Barrow in his method are, of course, 
logically to be justified only in terms of limits. Barrow undoubtedly 
thought of the triangle MRN as rigorously congruent to MTP only 
when it was infinitesimally small, for he said here and elsewhere, that 
"if the arc MN is assumed to be indefinitely small, we may safely sub- 
stitute instead of it the small bit of the tangent." 301 Passages such as 
these reappear in the work both of Newton and Leibniz, showing how 
difficult it was at that time for mathematicians to think arithmetically 
in terms of limits. 

Of all the mathematicians who anticipated portions of the differ- 
ential and the integral calculus, none approached more closely to the 
new analysis than did Fermat and Barrow. The former invented 
analytic methods of procedure equivalent both to differentiations and 
to integrations, ,but he appears not to have realized fully the signifi- 
cance of the interrelation between the two types. On the other hand, 
Barrow appears to have discovered the fundamental inverse relation- 
ship, but because he did not develop fully the possibilities of the ana- 
lytic representation of the operations involved, he was unable to make 
effective use of it. 302 He systematically reduced inverse-tangent prob- 

300 More, Isaac Newton, p. 185, n. 301 Geometrical Lectures, p. 61 ; cf. also pp. 120-21. 

302 See Zeuthen, "Notes," 1897, pp. 565-606; or his Gesckichte der Malhematik itn XVI. 
una 1 XVII. Jahrhundert, pp. 345-57, for excellent analyses of this portion of Barrow's 
work. 



A Century of Anticipation 185 

lems to quadratures, but he did not convert the latter, by means of his 
inversion theorem, into considerations derived from tangent deter- 
minations — that is, he did not express them in terms of the anti- 
derivative, as is generally done in the calculus. Barrow saw no ad- 
vantage in doing this, because he had not reduced his tangent method 
to a simple algorithmic form, as did Newton and Leibniz shortly 
thereafter. Had he done so, he would without doubt have forestalled 
these men as the founder of the calculus. 303 

This, however, Barrow did not even attempt to do. After preparing 
his lectures for publication, he turned them over to Newton and Collins 
for final revision, and gave up mathematics for the study of theology. 
Newton displayed the analytical knowledge required to develop the 
geometrical ideas of Barrow into an algorithm, and in fact was already 
in possession of his methods of the calculus several years before 1670, 
the year in which Barrow's Geometrical Lectures were published. In 
fact, there had appeared earlier in the Low Countries a number of rules 
which in application so closely resemble parts of the calculus that they 
may be thought of as forming a transition from the infinitesimal pro- 
cedures, developed during this century of anticipation, to the methods 
of fluxions and differentials. Such rules for tangents and for maxima 
and minima as those formulated by Sluse, Hudde, and Huygens did 
not involve new basic conceptions. They were simply canonical forms 
derived from earlier methods, particularly those of Fermat and Barrow. 

Sluse, for example, was perhaps the first mathematician to give a 
general algorithm for writing down, without following through the 
analytic calculation required by such methods as those of Barrow and 
Fermat, the tangent to a curve whose equation is rational in x and 
y. 30i His rule, which he seems to have formulated in 1652 306 but which 
was not published until 1673, may be stated in the following terms. Let 
the equation be f(x } y) = 0. Then the sub tangent will be the quotient 
obtained by placing in the numerator all the terms containing y, each 
multiplied by the exponent of the power of y appearing in it; and in the 
denominator all the terms containing x, each multiplied by the ex- 

** Child would categorically attribute the invention to Barrow, because, using modern 
analytical notations instead of the synthetic form of the original, he reads into this work 
algebraic methods of procedure, rather than geometric proofs of propositions. 

"* Rosenfeld, "Ren6 Francois de Sluse et le probleme des tangentes." 

** Le Paige, "Correspondance de Rene" Francois de Sluse." 



186 A Century of Anticipation 

ponent of the power of x appearing in it and then divided by x. m 

. v f (x y) 

This is, of course, equivalent to forming the quotient ' y ' , but 

/»(*, y) 

Sluse was not thinking in terms of derivatives of functions, as we do. 
He did not give a general demonstration of his rule, but such is easily 
supplied by the methods of either Fermat or Barrow. 

Johann Hudde, in 1659, gave an exactly analogous rule, 307 and this 
was rediscovered a few years later by Christiaan Huygens. 308 Hudde 
stated also a rule for writing down, without carrying out the work of 
Fermat's method, the maximum or minimum value of a rational func- 
tion of one variable. 309 This is equivalent to setting equal to zero the 
derivative of the function as found by the rule for quotients given in 
calculus textbooks. 

The almost simultaneous appearance of such rules and formulas in- 
dicates that shortly after the middle of the seventeenth century in- 
finitesimal considerations were so widely employed and had developed 
to such a point that, given a suitable notation, a unifying analytic 
algorithm was almost bound to follow. Even Huygens, who in his 
earlier work had scrupulously followed the methods of the ancients, 
was after 1655 au courant with the new points of view and made fre- 
quent use of them. He attempted a generalization of the methods of 
the ancients, in the manner of Valerio; 310 he repeated the infinitesimal 
demonstrations of Galileo and Torricelli on falling bodies; 311 he showed 
the influence of Cavalieri in speaking of lines as elements of surfaces ; 312 
and his work includes frequent applications of Fermat's methods of 
tangents and maxima and minima. 313 Huygens, however, was a mathe- 
matical classicist, and it remained for his two younger friends, Newton 
and Leibniz, to bind all of this work into what represents probably the 
most effective instrument for scientific investigation that mathematics 
has ever produced. 314 

804 Sluse, "A Method of Drawing Tangents to All Geometrical Curves," p. 38. 
m Huygens, (Euvres completes, XIV, 446-47. 
*» Ibid., XIV, 442-48, 504-17. 

** See the second letter by Hudde in Geometria, a Renoto des Cartes anno 1637 Gattice 
edita, I, 507-16. 
810 (Euvres completes, XIV, 338-39. m Ibid., XVI, 114-18. 

m Ibid., XI, 158; XII, 5; XIII, 753; XIV, 192, 337. 
m Ibid., XI, 19; XVI, 153; XIV, passim. 
314 Cf. Bell, The Queen of the Sciences, p. 8. 



V. Newton and Leibniz 



FEW NEW branches of mathematics are the work of single 
individuals. The analytic geometry of Descartes and Fermat was 
certainly not the result of their investigations only, but was the 
outgrowth of several mathematical trends which converged in the 
sixteenth and seventeenth centuries. It was the result of the influence 
of Apollonius, Oresme, Viete, and many others. 

Far less is the development of the calculus to be ascribed to one or 
two men. We have followed the long and uneven flow of thought which 
led from the philosophical speculations and the mathematical demon- 
strations of the ancients to the remarkably successful heuristic 
methods of the seventeenth century. We have indicated that the 
procedures invented by Fermat, for example, are almost identical 
with those found in the calculus, and that the new propositions dis- 
covered by Barrow include the geometrical equivalent of the basic 
theorem of the subject. 

The time was indeed ripe, in the second half of the seventeenth 
century, for someone to organize the views, methods, and discoveries 
involved in the infinitesimal analysis into a new subject characterized 
by a distinctive method of procedure. Fermat had not done this, 
largely because of his failure to generalize his methods and to recog- 
nize that the problems of tangents and quadratures were two aspects 
of a single mathematical analysis — that the one was the inverse of 
the other. Barrow was unable to establish the new subject for, al- 
though the first to recognize clearly the unifying significance of this 
inverse property, 1 he failed to realize that his theorems were the basis 
for a new subject. Being unsympathetic with the Cartesian mathe- 
matical analysis and the algebraic trend, he implied that his results 
were to be considered as rounding out the geometry of the ancients. 2 

The traditional view, therefore, ascribes the invention of the cal- 
culus to the more famous mathematicians, Isaac Newton and Gott- 
fried Wilhelm von Leibniz. From the point of view of the development 
of the concepts involved, the aspect which concerns us chiefly here, 

1 Geometrical Lectures, p. 124. * Ibid., p. 66. 



188 Newton and Leibniz 

it might be far better to speak of the evolution of the calculus. Never- 
theless, inasmuch as Newton and Leibniz, apparently independently, 
invented algorithmic procedures which were universally applicable 
and which were essentially the same as those employed at the present 
time in the calculus, and since such methods were necessary for the 
later logical development of the conceptions of the derivative and the 
integral, there will be no inconsistency involved in thinking of these 
men as the inventors of the subject. In doing so, however, we are not 
to consider or to imply that they are responsible for the ideas and defi- 
nitions underlying the subject at the present time; for these basic 
notions were to be rigorously elaborated only after two centuries of 
further effort in this direction. Furthermore, inasmuch as we are here 
more concerned with ideas than with rules of procedure, we shall not 
discuss the shamefully bitter controversy 3 as to the priority and 
independence of the inventions by Newton and Leibniz. 4 Both men 
owed a very great deal to their immediate predecessors in the develop- 
ment of the new analysis, and the resulting formulations of Newton 
and Leibniz were most probably the results of a common anterior, 
rather than a reciprocal coincident, influence. 

Attempts have been made by historians of the calculus 5 to trace 
two distinctly different threads of development: one, the kinematic, 
leading to Newton through Plato, Archimedes, Galileo, Cavalieri, and 
Barrow; and the other, the atomistic, tending toward Leibniz through 
Democritus, Kepler, Fermat, Pascal, and Huygens. There is, how- 
ever, a complete lack of recognition of such a cleavage by the mathe- 
maticians involved, nor can we now distinguish the views and methods 
of the one "group," throughout the seventeenth century, from those 

1 See Hathaway, "The Discovery of Calculus," pp. 41-43; and "Further History of the 
Calculus," pp. 166-67, 464-65, for accusations against Leibniz of "the foundation of a plot 
to deprive Newton of all credit . . . , with typical German propaganda," and of inaugu- 
rating "that system of espionage on scientific work in foreign countries by which the 
usefulness and credit of as much of that work as possible might be transferred to Germany." 

4 See Sloman, The Claim of Leibnitz to the Invention of the Differential Calculus; Leibniz, 
The Early Mathematical Manuscripts (ed. by Child) for the statement of the suspicions 
directed against Leibniz. For his defense see Gerhardt's two works, Die Entdeckung der 
Differ entialrechnung durch Leibniz, and Die Entdeckung der hoheren Analysis; and also 
Mahnke's two articles, "Neue Einblicke in die Entdeckungsgeschichte der hoheren 
Analysis," and "Zur Keimesgeschichte der Leibnizschen Differentialrechnung." For an 
extensive chronological bibliography on the subject of the controversy, see De Morgan, 
Essays on the Life and Work of Newton. 

6 See, e. g., Hoppe, "Zur Geschichte der Infinitesimalrechnung," pp. 175-76. 



Newton and Leibniz 189 

of the other. Galileo, Cavalieri, Torricelli, and Barrow used both 
fluxionary and infinitesimal considerations, and the procedures of 
Fermat, Pascal, and Huygens were perhaps as well known to Newton 
and the English mathematicians as to Leibniz. The geometric devel- 
opments leading to the fluxionary calculus of Newton were not es- 
sentially other than those pointing the way toward the differential 
calculus of Leibniz. However, after the methods of procedure of the 
subject had been established, and the logical and metaphysical bases 
of these were brought into question, the contrast between the points 
of view and the modes of presentation became heightened by the 
contrasting scientific and philosophic tastes of the inventors, as well, 
perhaps, as by the priority controversy in which blind loyalties 
prevented their successors from appreciating the advantages and dis- 
advantages of the two systems. We shall attempt, therefore, to point 
out here not only the origins of the work of Newton and Leibniz, but 
also the nature of the interpretations which they later gave and the 
significance of these for the development of the fundamental notions 
of the calculus. 

Isaac Newton was the student, at Cambridge, of Isaac Barrow and 
so came strongly under the influence of the latter, whose Geometrical 
Lectures he helped to prepare for publication. Now JBarrow jyas 
familiar with the work of Cavalieri and the two views of the genera- 
tion of geometrical magnitudes there presented — that of indivisibles, 
and that of flowing quantities. He thought of a tangent to a curve not 
only as the prolongation of one of the infinitely many lineal elements 
of which the curve might be assumed to be composed, but also as the 
direction of motion of a point which, by moving, generated the curve. 
These views were almost certainly familiar to Newton through his 
attendance at Barrow's lectures. Barrow, however, lacked an apprecia- 
tion of the analytical methods of Descartes and Fermat, and failed 
to realize the significance of Wallis' arithmetization. These newer 
views, however, had been presented in 1655 by the works of Wallis, 
to which we have already referred and with which Newton became 
acquainted during the period of his early mathematical training. 6 
Newton, in fact, acknowledged that he had been led to his first dis- 
coveries in analysis and fluxions by the Arithmetica infinitorum of 

6 Sloman, The Claim of Leibnitz to the Invention of the Calculus, pp. 1-7. 



190 Newton and Leibniz 

Wallis, 7 and the principles of induction and interpolation which 
Wallis there employed may have been instrumental also in leading 
Newton to the discovery of the binomial theorem. 8 Newton's concep- 
tion of number resembles that of Wallis rather than"that of Barrow- 
less a collection of units than an abstract ratio of any quantity to 
another, a definition which also includes irrational ratios as numbers. 9 
Newton in this respect went beyond Wallis and Descartes in regarding 
negative ratios as numbers in the true sense of the word 10 — a gen- 
eralization of the geometrical representation of Descartes. 

Another element adding to the effectiveness of Newton's presenta- 
tion of the method of fluxions was the use of infinite series. The 
Scholastic philosophers had studied infinite series in connection with 
the geometric representation of variability, and Gregory of St. Vin- 
cent, Tacquet, and Fermat had made use of infinite progressions. 
However, the earliest investigations of general arithmetic infinite 
series were largely the work of English mathematicians such as 
Wallis and James Gregory. Incidentally, the important work in this 
connection by the latter appeared only a year or two before Newton 
composed his first treatise on the calculus — one in which he employed 
infinite series in connection with the binomial theorem. The use of 
such series did indeed make for a universality of application of the 
method of fluxions and aided in freeing it from geometrical prejudices, 
but there has been a tendency on the part of historians to focus 
attention upon Newton's use of infinite series, rather than upon other 
more essential aspects of his work. 11 
/ Newton tells us that he was in possession of his fluxionary calculus 
\ as early as 1665-66, 12 that is, at some time during the period in which 
he had heard Barrow's lectures and had discovered the binomial the- 
orem. The first notice of his calculus was given, however, in 1669, in 
De analyst per aequationes numero terminorum infinitas. 13 This was 
not published until 1711, but it circulated among his friends. In this 

7 More, Isaac Newton, p. 184. 

8 Merton, "Science, Technology and Society in Seventeenth Century England," p. 
472, n. 

8 Newton, Opera omnia, I, 2. 

10 Ibid., I, 3. Cf. Schubert, "Principes fondamentaux de l'arithmetique," pp. 35-37. 

11 Cf. Zeuthen, "Notes," 1895, pp. 194 ff., and a review of these by Tannery in Bulletin 
de Darboux, 2d series, XX (1896), 24^28. 

12 Opera omnia, I, 333. M Opera omnia, I, 257-82; Opuscula, I, 3-28. 



Newton and Leibniz 191 

monograph he did not explicitly make use of the fluxionary notation 
or idea. Instead, he used the infinitely small, both geometrically and 
analytically, in a manner similar to that found in Barrow and Fermat, 
and extended its applicability by the use of the binomial theorem. In 
this paper Newton employed the idea of an indefinitely small rectangle 
or "moment" of area and found the quadratures of curves as follows: 
Let the curve be so drawn that for the abscissa x and the ordinate y 

(\ m +n 
1 ax n . Let the moment or infinitesimal 
m + nj 

increase in the abscissa, following the notation of James Gregory, 

be o. The new abscissa will then be x + o and the augmented area 

I n \ ' !L±Jt 

z + oy = ( I a(x + o) n . If in this expression we apply the 

\m + n) 

binomial theorem, divide throughout by o, and then neglect the 

m 

terms still containing o, the result will be y = ax n . That is, if the 

^ m + n *» 

ft - 

area is given by z = ax n , the curve will be y = ax n . 

m -f- n 

- n ^^ 

Conversely, if the curve is y = ax n , the area will be z = ax * , li 

m-\-n 

Here we have an expression for area which was arrived at, not 

through the determination of the sum of infinitesimal areas, nor 

through equivalent methods which had been employed by Newton's 

predecessors from Antiphon to Pascal. Instead, it was obtained by a 

consideration of the momentary increase in the area at the point in 

question. In other words, whereas previous quadratures had been 

found by means of the equivalent of the definite integral defined as 

a limit of a sum, Newton here determined first the rate of change of 

the area, and then from this found the area itself by what we should 

now call the indefinite integral of the function representing the 

ordinate. It is to be noted, furthermore, that the process which is 

made fundamental in this proposition is the determination of rates 

of change. In other words, what we should now call the derivative 

is taken as the basic idea and the integral is defined in terms of this. 

Mathematicians from the time of Torricelli to Barrow had in a sense 

known of such a relationship, but Newton was the first man to give 

14 Opera omnia, I, 281 ; Opuscula, I, 26. 



192 Newton and Leibniz 

a generally applicable procedure for determining an instantaneous rate 
of change and to invert this in the case of problems involving sum- 
mations. Before this time the tendency had been rather in the opposite 
direction — to reduce problems, whenever possible, to the determina- 
tion of quadratures. With this step made by Newton, we may consider 
that the calculus has been introduced. 
Newton applied this method to the quadrature of numerous curves, 

f a 2 

such as y — x 2 -f- x and y = . A few years later, in sending 

b -f- x 

these results to Collins, he described in addition a number of proposi- 
tions in maxima and minima and in tangents, which he had obtained 
by means of his methods. It is this letter to Collins of December 10, 
1672, which became significant in the controversy as to whether 
Leibniz made his discoveries independently of Newton. In this letter 
Newton pointed out frankly that his rules are analogous to those of 
Sluze and Hudde, although more general; 16 and in another place he 
admitted that he had gotten the hint for his procedure from the 
method of Fermat, improved by Gregory and Barrow. 16 The change 
from the E of Fermat to the o of Gregory and Newton is, of course, 
trivial. It has from time to time been interpreted as a substitution of 
zero for E, a view which would reduce Newton's method to a mean- 
ingless manipulation of zeroes, somewhat in the manner of Bhaskara. 17 
Newton distinctly regarded his symbol as the letter o and not the 
cipher zero, and in this respect it is entirely comparable to Fermat's 
E. The significance of Newton's work lay first of all in the fact that 
he applied the method "directly and invertedly," as he said. 18 In the 
second place Newton regarded it, in connection with the use of infinite 
series, as a universal algorithm, whereas that of Fermat, as well as the 
modifications of this by Sluse, Hudde, and Huygens, availed only in 
the case of rational algebraic functions. 

It will be noticed that although the work of Newton contains the 
essential procedures of the calculus, the justification of these is not 

16 Opera omnia, IV, 510; cf. also Mathematical Principles (Cajori), pp. 251-52. 
14 More, Isaac Newton, p. 185, n. 

17 The entire interpretation of Newton given by Hoppe, "Zur Geschichte der Infinitesi- 
malrechnung," is vitiated by the fact that he has followed certain older historians in this 
mistake. Cf. Gerhardt, Die Entdeckung der hbheren Analysis, p. 80; Weissenborn, Die 
Principien der hbheren Analysis, p. 25, n.; Gerhardt, "Zur Geschichte des Streites," p. 131. 

u More, loc. cit. 



Newton and Leibniz 193 

clear from the explanation he gave. Newton did not point out by what 
right the terms involving powers of o were to be dropped out of the 
calculation, any more than Fermat justified omitting the powers of 
E, or Barrow those of e and a. His contribution was that of facilitating 
the operations, rather than of clarifying the conceptions. As Newton 
himself admitted in this work, his method is "shortly explained 
rather than accurately demonstrated." 

m 

In his demonstration above that the area of y = ax* is given by 

n "-^ 

z = ax n , however, we can see some hint of the thought in 

n + m 

his mind. The ordinate y seems to represent the velocity of the increas- 
ing area, and the abscissa represents the time. Now the product of the 
ordinate by a small interval of the base will give a small portion of the 
area, and the total area under the curve is only the sum of all of these 
moments of area. This is exactly the infinitesimal conception of Oresme, 
Galileo, Descartes, and others, in their demonstrations of the law of 
falling bodies, except that these men had found the area as a whole 
through the addition of such elements, whereas Newton found the 
area from its rate of change at a single point. It is difficult to tell in 
exactly what manner Newton thought of this instantaneous rate of 
change, but he very likely accepted it as similar to the conception of 
velocity which Galileo had made so familiar but had not defined 
rigorously. A thorough-going empiricist for whom mathematics was a 
method rather than an explanation, 19 Newton apparently considered 
any attempt to question the instantaneity of motion as linked with 
metaphysics, and so avoided framing a definition of it. Nevertheless, 
he accepted this notion and made it the basis of his second and more 
extensive exposition of the calculus, as given in the Methodus fluxionutn 
et serierum infinitarum, 20 which was written about 167 1, 21 but not 
published until 1736. 

In this book Newton introduced his characteristic notation and 
conceptions. Here he regarded his variable quantities as generated by 
the continuous motion of points, lines, and planes, rather than as 
aggregates of infinitesimal elements, the view which had appeared in 

19 Burtt, Metaphysical Foundations, pp. 208-10. 

20 Opusctda, I, 31-200. 

21 See Zeuthen, "Notes," 1895, p. 203; cf. also Newton, Opera omnia, II, 280. 



194 Newton and Leibniz 

De analyst. Just as Barrow found the chief characteristic of time in 
its even flow, so also his pupil Newton, although he did not "formally 
consider time," 22 was influenced to make continuous motion funda- 
mental in his system. This concept Newton seems to have felt was 
sufficiently impelling and so clearly known through intuition as to 
make further definition unnecessary. The rate of generation Newton 
called a fluxion, designating it by means of a letter with a dot over it, 
a "pricked letter"; the quantity generated he called a. fluent, employ- 
ing in this connection the terms which had appeared earlier in the 
work of Calculator. Thus if x and y are the fluents, then their fluxions 
are x and y. Incidentally, Newton in other places 23 proceeded to point 
out that one may consider the fluxions x and y, in turn, as fluents of 
which the fluxions are represented by 'x and y, and so on. The fluents 
of which x and y are the fluxions Newton represented by x and y; 
the fluents of which these latter quantities are the fluxions were 
written x and y, and so on. 

In the Methodus fluxionum Newton stated clearly the fundamental 
problem of the calculus: the relation of quantities being given, to 
find the relation of the fluxions of these; and conversely. 24 In con- 
formity with this problem and the new notation, Newton then gave 
examples of his method. These may be represented by the determina- 
tion of the fluxion of y = x 11 . His approach in this case is but slightly 
different from the earlier exposition in De analyst. If o is an infinitely 
small interval of time, then xo and yo will be the indefinitely small 
increments, or moments, of the flowing quantities x and y. In y = x n 
one then substitutes x + xo for x and y -f- yo for y, expands as before 
by the binomial theorem, cancels the terms not containing o, and 
divides throughout by o. Since, moreover, o was assumed to be 
infinitely small, the terms containing this — that is, the moments of 
quantities — can be considered as zero in comparison with the others, 
and are to be neglected. 28 

The result, y = nx n ~ i x, is, of course, the same as that obtained 
by Newton previously in the De analyst without the use of fluxions. 

22 Opuscula, I, 54. 

23 Cf. Opera omnia, I, 338; Opuscula, I, 208. M Opuscula, I, 55, 61. 

26 Opuscula, I, 60. The illustration, y = x", which we have presented is not here given 
in this form by Newton, but is taken as representative, in form and argument, of the 
exercises he gave. 



Newton and Leibniz 195 

It is to be noted that the introduction here of the conception of a fluxion 
is not an essential modification of the earlier work. The infinitely small 
enters as persistently as in the 1669 exposition, but in the dynamic 
form of Galileo's moment or the conatus of Hobbes rather than in 
the static form of Cavalieri's indivisible. This change serves only in 
intuition to remove the harshness (as Newton expressed it) from the 
doctrine of indivisibles. 26 In thought the justification of the neglect 
of infinitely small terms is to be made on precisely the same basis, 
whether it be written E, e, a, o, or ox. Newton himself seemed to feel 
here some need for the limit concept, for he pointed out that fluxions 
are never considered alone, but always in ratios. 27 Later, when Newton 
sought escape from the clutches of the infinitely small, he emphasized 
this fact much more strongly. 

This third stage in his thought appears clearly in De quadratura 
curvarum, 28 which was written in 1676 but not published until 1704. 
In this treatise Newton sought to remove all traces of the infinitely 
small. Mathematical quantities were not to be considered as made 
up of moments or very small parts, but as described by continuous 
motion. In determining the fluxion of x n , Newton proceeded much as 
in the Methodus fluxionum, replacing x by (x + o). In conformity 
with the fluxionary symbolism it would be expected that the incre- 
ment in x should be designated ox instead of o, but inasmuch as 
Newton is here dealing with only a single variable, the fluxion of this 
may conveniently be taken as unity. On expanding (x + o) n by the 
binomial theorem, and subtracting x n , the result is, of course, the 
change in x n corresponding to the change o in x. Instead, now, of 
completing the argument by a doubtfully justified neglect of terms, 
Newton formed the ratio of the change in x to the change in x n : 

that is, 1 to nx n ~ 1 + n I J ox n ~ 2 + . . . ; and in this he allowed 

o to approach zero — to vanish. The resulting ratio, 1 to nx n ~ *, we 
should speak of as the limit of the ratio of the changes, but Newton 
called it the ultimate ratio of the changes — a terminology which 
was later to lead to some confusion in thought. This ultimate ratio 
of "evanescent increments" is the same as the prime or first ratio of 

28 Opera omnia, I, 250; II, 39. 

27 Opuscula, I, 63-64. M Opera omnia, I, 333-86; Opusculo, I, 203-44. 



196 Newton and Leibniz 

the "nascent augments." It is likewise the ratio of the fluxions at the 
point in question. 29 

In the above demonstration, the essential elements of the derivative 
are more clearly present than in any other part of Newton's work: 
the emphasis upon a function of one variable, rather than upon an 
equation in several; the formation of the ratio of the changes in the 
independent variable and in the function; and, finally, the deter- 
mination of the limit of this ratio as the changes approach zero. 
Incidentally, the ratio as expressed by Newton is commonly inverted 
in the modern derivative. There are in Newton's thought, moreover, 
certain elements which have since been discarded as adscititious: his 
appeal to time as an auxiliary independent variable is now considered 
gratuitous; and the limiting ratio is now regarded as a single number, 
rather than as the quotient of two rates of change. Had Newton 
devoted more of his time to clarifying the elements of thought in his 
demonstration by ultimate ratios, the calculus might have been 
established upon the concept of the derivative a century before the 
time of Cauchy. In his first published account of his new analysis 
Newton suggested this type of argument; but in his illustrations of 
the method of fluxions in this work he unfortunately resorted to the 
infinitesimal terminology of his earlier accounts. 

Newton's discovery of the calculus dates back to the years 1665 
and 1666, as he says in De quadrature/,. Within the following decade 
he had written out, as we have seen, three accounts of his methods, 
but had published nothing. By 1676 he became aware that Leibniz 
was working on similar problems, and on October 24 of this year 
he sent a letter to Leibniz, through Oldenburg, in which he gave in 
the form of an anagram a statement of the fundamental problem 
of his calculus. This seems to have been his only effort to assure his 
claim to priority in the invention of the calculus. Upon transposing 
the letters of this anagram and translating, it read: "Given in an 
equation the fluents of any number of quantities, to find the fluxions 
and vice versa." 30 Similar statements of the problem of the calculus 
had been included in the Methodus fluxionum and the De quadratura 
which he had already composed. 31 

" Opera omnia, 1, 334. 

n Opera omnia, IV, 540 ff.; cf. also Leibniz, Mathematische Sckrifkn, I, 122-47. 

u Opera omnia, I, 339, 342; Opuscule, I, 55, 61. 



Newton and Leibniz 197 

In this letter he also admitted his indebtedness to Wallis, James 
Gregory, Sluse, and others, but did not give an exposition of his 
methods. The first published account of his calculus appeared some- 
what incidentally, more than ten years later, in the famous Principia 
mathematica philosophiae naturalis of 1687. The propositions in this 
book, concerned as they are with velocities, accelerations, tangents, and 
curvatures, are largely those handled now by the methods of the 
calculus, but Newton presented them in the form of synthetic geo- 
metrical demonstrations with an almost complete lack of analytical 
calculations. Nevertheless, Newton at several points in the work gave 
indications of more general points of view. 

In a series of lemmas in the first book, he expressed the type of 
argument appearing in De quadratura curvarum. "Quantities, and the 
ratios of quantities, which in any finite time converge continually to 
equality, and before the end of that time approach nearer to each 
other than by any given difference, become ultimately equal." 32 This, 
of course, is the sort of general limit proposition which Stevin, Valerio, 
Gregory of St. Vincent, Tacquet, Wallis, and others had attempted to 
substitute for the Greek method of exhaustion. In fact a passage in 
Gregory of St. Vincent, 33 in which the word "terminus" was used to 
designate the limit of a progression, may have been the origin of the 
term "ultimate ratio" which Newton was to use so frequently. New- 
ton's view of a limit, like that of these earlier workers, was bound 
up with geometric intuitions which led him to make vague and 
ambiguous statements. Thus he said, "The ultimate ratio of the arc, 
chord, and tangent, any one to any other, is the ratio of equality." 34 
and a little later he spoke of the similarity of the "ultimate forms of 
evanescent triangles." 36 

These remarks imply that Newton was not thinking arithmetically, 
as we do now, of the limit of the sequence of numbers representing 
the ratios of the (arithmetical) lengths of the geometrical quantities 
involved, as these become indefinitely small, but that he also was 
influenced by the infinitesimal views of the seventeenth century to 
think of ultimate geometrical indivisibles. It is true that he never 
used the expressions ultimate arcs, chord, tangents, or triangles, but 
only those of ultimate ratios and forms, expressions which allow of 

M Opera omnia, I, 237; II, 30. M Opus geometricum, p. 55. 

« Opera omnia, I, 242; II, 34. » Ibid., I, 243. 



198 Newton and Leibniz 

rigorously correct abstract interpretations, but which strongly sug- 
gest others in terms of the intuitively more attractive view afforded 
by infinitesimals. That Newton realized the difficulties involved in a 
naive view of infinitesimals is indicated, however, in his further 
statement in the Principia that "Ultimate ratios in which quantities 
vanish, are not, strictly speaking, ratios of ultimate quantities, but 
limits to which the ratios of these quantities decreasing without limit, 
approach, and which, though they can come nearer than any given 
difference whatever, they can neither pass over nor attain before the 
quantities have diminished indefinitely." 36 This is the clearest state- 
ment Newton gave as to the nature of ultimate ratios, but we shall 
find that, in continuing this argument in a lemma in the second book 
of his Principia, his exposition again took on more strongly the 
dependence upon the idea of infinitely small quantities, with the con- 
cept of limit somewhat hazily implied as basic. It is precisely this 
lack of arithmetical clarity which led, in the following century, to 
controversial discussions, not only on the validity of Newton's flux- 
ions, but also as to what Newton really meant by the above statements 
and others similar to them. 

Inasmuch as the Principia is written in the old synthetic geometric 
manner, references to the method of fluxions are not numerous. In 
the second book, however, there appeared the first publication of 
"the foundation of that general method." 37 Here one finds the state- 
ment of the fundamental principle, "The moment of any genitum is 
equal to the moments of each of the generating sides multiplied by 
the indices of the powers of those sides, and by their coefficients con- 
tinually." Newton proved this first for the product AB as follows: 
Let AB represent a rectangle and let the sides A and B be diminished 
by \a and \b respectively. The diminished area will then be AB — 
\aB — \bA -f- \ab. Now let the sides of AB be increased by \a and 
\b respectively. The area of the enlarged rectangle will then be AB 
-f- \aB -f \bA + \ab. Subtracting the smallest rectangle from the 
largest, one obtains aB -f bA as the moment of the original rectangle, 
corresponding to the moments a and b of A and B; which proves the 
proposition for this product. If A = B, the moment of A 2 is deter- 
mined in turn as 2aA . 

*• Opera omnia, I, 251. » Opera omnia, II, 277-80. 



Newton and Leibniz 199 

By the use of the decrements \a and \b and the increments \a 
and \b, instead of the increments a and b, Newton here avoided the 
necessity of dropping the infinitely small term ab. Newton thus made 
explicit use of infinitely small quantities of first order only — in which 
respect his work is to be contrasted with that of Leibniz — but his 
procedure in the proposition above was later justly criticized as 
implying the omission of infinitesimals of second order. 

To find the moment of ABC Newton let AB = G and, by applying 
the first part of the theorem, obtained cAB + bCA + aBC as the 
result. Letting A = B = Cj the moment of A 3 is in turn determined 
as 3aA 2 . By similar procedures the moment of A" for positive integral 
powers is found to be naA n ~ l . This same result is seen to hold for 
negative powers also. This is apparent from the following considera- 
tions. Let m be the moment of — . Then from — - A = \ and the 

A A 

1 a 

moment of a product, one obtains -•c + i.w = 0,orm= — . 

A A 2 

This argument is easily generalized to include all negative integral 

powers, and, with slight modifications, it is applicable to all*products 

of rational powers of variables. 

Newton said 38 that this is the foundation of his method of tangents 
and quadratures; and, in fact, this rule, combined with the use of 
infinite series, is sufficient for arriving at the essential results of the 
method of fluxions. However, because Newton supplied here such an 
unfortunately brief exposition of his procedure and its justification, 
and inserted this short account; in . the second book in the un- 
obtrusive form of a lemma 39 to oflier propositions, some doubt 
has been expressed as to the seriousness with which it was put 
forward. 40 

The basis of the calculus as thus first published in the Principia, 
is, of course, to be found in the nature of Newton's moments; but 
it is just here that Newton was very far from clear in his language. 
He said on this point, "Finite particles are not moments, but the 

88 Ibid. » Opera omnia, II, 277-80. 

"Moritz Cantor (Vorlesungen, III, 192) would put but little emphasis upon the 
lemmas in the Principia, whereas Zeuthen ("Notes," 1895, p. 249; cf. also Geschichte der 
Mathematik im XVI. and XVII. Jahrhundert, pp. 382-84) stresses their significance in the 
calculus. 



200 Newton and Leibniz 

very quantities generated by the moments. We are to conceive them 
as the just nascent principles of finite magnitudes." Perhaps realizing 
that this statement made his moments as vague as the infinitesimals 
of Cavalieri, Fermat, and Barrow, he justified himself by adding, 
"Nor do we in this Lemma regard the magnitude of the moments, 
but their first proportion as nascent." 41 This looks like an attempt 
to bring in the doctrine of limits, which he had formulated in Book I, 
in which he regarded the ratio as ultimate, without specifying that 
the quantities entering it were so. Nevertheless, it is difficult to see 
just how one is to think of the limit of a ratio in determining the 
moment of AB. We have to deal here with two variables and are faced 
with the equivalent of partial differentiation, unless we have recourse 
to time as a single independent variable, as Newton next suggested. 
Perhaps realizing the difficulties in the way of interpreting the proposi- 
tion in terms of the ratio or proportion of infinitesimals, Newton added 
another interpretation. "It will be the same thing, if, instead of 
moments, we use either the velocities of the increments and decre- 
ments (which may also be called the motions, mutations, and fluxions 
of quantities), or any finite quantities proportional to those veloc- 
ities." 42 

To summarize the above, we see that Newton first had in mind 
infinitely small quantities which are not finite nor yet precisely zero. 
"Ghosts of departed quantities" they were fittingly called by the 
critics of the method in the following century. These offer too great 
difficulty of conception, so Newton next focused attention on their 
ratio, which in general is a finite number. Knowing this ratio, one 
may now substitute for the infinitesimal quantities forming it any 
other easily conceived finite magnitudes having the same ratio, such 
as quantities which are thought of as the velocities or fluxions of 
those entering into the equation. Newton thus offered in the Principia 
three modes of interpretation of the new analysis: that in terms of 
infinitesimals (used in his De analysi, his first work) ; that in terms of 
prime and ultimate ratios or limits (given particularly in De quad- 
ratures, and the view which he seems to have considered most rigor- 
ous) ; and that in terms of fluxions (given in his Methodus fluxionum, 
and the one which appears to have appealed most strongly to his 

41 Opera omnia, II, 278. <* Ibid. 



Newton and Leibniz 201 

imagination). The fact that Newton could thus present all three views 
as essentially equivalent shows us how far he was from viewing his 
method as quite distinct from the somewhat equivalent methods of 
his predecessors and contemporaries. In De guadratura, after saying 
that he there considered quantities as described by a continued 
motion and used the method of prime and ultimate ratios, he asserted 
that his method is consonant to the geometry of the ancients; 43 and 
in the Principia he also admitted that Leibniz had a similar method 
for considering the generation of magnitudes — an admission which 
was, however, omitted from later editions. 44 In fact the method of 
fluxions is dependent upon some other method, such as limits or 
infinitesimals, for the determination of the basic relations between 
the fluxions. Although Newton apparently preferred to link his 
method of fluxions with the idea of a limiting ratio, he so often used 
infinitesimals for dispatch that we shall find many of his successors 
later interpreting the fluxions themselves as infinitely small quan- 
tities, confusing them with moments. 

Newton himself frequently used the concept of the infinitely small 
throughout his early work, but tended to become wary of it in his later 
expositions. In a portion of De guadratura which appeared in Wallis's 
Algebra of 1693, Newton had said that terms multiplied by o he omitted 
as infinitely small, thus obtaining the result. 46 In the 1704 publication 
of the work, on the other hand, he said clearly, that "errors are not 
to be disregarded in mathematics, no matter how small." 46 The con- 
clusion was to be reached, not by simply neglecting infinitely small 
terms, but by finding the ultimate ratio as these terms become evan- 
escent. However, even after this he did not abjure the infinitesimal 
completely, but continued to speak of moments as infinitely little 
parts. Furthermore, Newton added to the confusion in the thought 
of his contemporaries on fluxions by failing sometimes to multiply the 
fluxions by o when he wished to represent moments. Although he said 
that wherever pricked letters represent moments and are without the 
letter o, this letter is always to be understood, very many English 
mathematicians began to associate fluxions with the infinitely small 

« Opera omnia, I, 338. ** Ibid., H, 280. 

« See De Morgan, "On the Early History of Infinitesimals in England," p. 324. See 
also Cajori, "Newton's Fluxions," p. 192, and Raphson, The History of Fluxions, p. 14. 
46 Opera omnia, I, 338. 



202 Newton and Leibniz 

differentials of Leibniz. 47 Newton's final view of the basis of the 
subject, however, seems to be that shown in his remark in De quad- 
rature "I have sought to demonstrate that in the method of fluxions 
it is not necessary to introduce into geometry infinitely small 
figures. 48 

We have seen that most of Newton's work on the calculus was 
written in the period from 1665 to 1676, but none of it had been pub- 
lished during that time. It has been suggested 49 that Newton's long 
delay in the publication of his three chief works on the calculus was 
occasioned by the fact that he was dissatisfied with the logical founda- 
tions of the subject. In the meantime, however, other mathematicians 
were looking for the general principle needed to solve the problems 
of tangents, maxima and minima, and quadratures. The methods of 
Fermat had already been modified by Huygens, Hudde, Sluze, and 
others. These men were contemporaries of the most versatile genius 
of the period, Gottfried Wilhelm von Leibniz, who, like Newton, was 
to develop rules and a literal symbolism for putting all of the infinites- 
imal considerations together under an algorithmic procedure. Although 
^interested primarily in law and logic, Leibniz had in his early days 
written a little on arithmetic and mechanics. In 1672, however, he 
met Huygens in Paris and was urged by him to make a deeper study 

/of mathematics. On a visit to London, in 1673, he met a number qL\- 
mathematicians^ learned much about infinite series, purchased a copy j 
of Barrow's Lectures, and may have/known, through Collins, of j 

\ Newton's De analyst. After his return to Paris in the same year, ^e ' 

studied the mathematical works ol Cavalieri, Torricelli, Gregory 

of St. Vincent, Roberval, Pascal, Descartes, Wren, James Gregory, 

Sluse, Hudde, and others. 60 The background of Leibniz in infinitesimal 

analysis was thus not greatly different from that of Newton; for the 

results of these men were well known in England as well as on the 

Continent. The early mathematical reading of Leibniz was thus largely 

on geometry, but he hapVother interests also which may have been 
y 

^Montucla (Histoire des mathematiques, II, 373) misinterprets JJewton's occasional 
omission of these letters as an indication of confusion in Newton's thought between 
velocity and increment. 

18 Opera omnia, I, 333; Opuscula, I, 203. 

49 Merz, A History of European Thought in the Nineteenth Century, II, 630. 

60 See Gerhardt, Die Entdeckung der Differentialrechnung durch Leibniz, p. 31 and 
passim. 



Newton and Leibniz 203 

decisive in shaping his analysis. His first mathematical paper had been 
on combinatorial analysis, and he always retained a strong arith- 
metical tendency. One of the first fruits of his study of problems in 
quadratures was the "Arithmetical Tetragonism," in which he found 
the area of a unit circle to be given by four times the infinite series 
1 _ i + £ — j. -j- ... .»i These formalistic and arithmetical con- 
siderations were now to combine in an interesting way with the 
geometry which Leibniz had begun to master. 

During his study at this time Leibniz was working on the problem 
of tangents, as well as that of quadratures, and had reached a solu- 
tion based upon the "characteristic triangle" — the differential triangle 
which had appeared in various forms, particularly in the works of 
Torricelli, Fermat, and Barrow. It is difficult to determine the filiation 
of events leading to the invention by Leibniz of the differential cal- 
culus, but he himself, in a letter written thirty years later, attributed 
the inspiration for his use of the differential triangle to a figure (fig. 18 
above) he had run across, about 1673, in Pascal's Traite des sinus du 
quart de cercle. 62 Leibniz said that on the reading of this example in 
Pascal a light suddenly burst upon him and that he then realized 
what Pascal had not — that the determination of the tangent to a 
curve depended upon the ratio of the differences in the ordinates and 
abscissas, as these became infinitely small, and that the quadrature 
depended upon the sum of the ordinates, or infinitely thin rectangles, 
for infinitesimal intervals in the abscissas. Moreover, the operations 
of summing and of finding differences were mutually inverse. Barrow 
had, in a sense, realized this also, for his a and e method for tangents 
involved the differences of ordinates and abscissas, his quadratures 
were effected by the summation of infinitesimals, and his inversion 
theorem showed the relationship of the two problems; but he had 
never developed these into a unified procedure. Leibniz, on the other 
hand, continued studying the characteristic triangle, encouraged by 

61 See Mathematische Sckriften, V, 88; cf. also Leibniz, Early Mathematical Manuscripts, 
p. 163. 

M Gerhardt and Zeuthen stress the obligation of Leibniz to Pascal. Child feels that 
inasmuch as Leibniz would have found in Barrow anticipations of the differential triangle 
much clearer than that in Pascal, he may have failed to mention Barrow either through a 
desire not to point out his indebtedness or through having forgotten the influence Barrow 
had had upon him. See especially Leibniz, Early Mathematical Manuscripts, pp. 15-16, 
for a discussion of the diagrams involved on this point; cf. also the works of Gerhardt and 
Zeuthen cited in the bibliography. 



204 



Newton and Leibniz 



Huygens, and related this work to his former interest in combinatorial 
analysis. 53 

In the harmonic triangle and in the arithmetic triangle of Pascal, 
there are striking relationships. For example, in the arithmetic tri- 

Arithmetic Triangle Harmonic Triangle 

111111... * 4 4 * 4 1 . . . 

12 3 4 5... * 

1 3 6 10 . . . 

1 4 10 . . . 

1 5 . . . 

1 . . . 



i 



i 



1 

■sir 



TV 



i 



*V 



TIT 



ifV 



7TT • 



angle any element is the sum of all of the terms in the line above it 
and to the left, and it is also the difference between the two terms 
directly below it. Similarly, in the harmonic triangle any element is 
the sum of the terms in the line below and to the right, and it is also 
the difference between the two terms just above it. 

That is, in Pascal's arithmetic triangle, if we designate by x the 
numbers in any one line, the numbers of the first line below will be 
the sum of all the x's up to this point, reading from left to right; 
those in the second line below will be the sum of the sums of all the 
x's, and so on. Conversely, the lines above represent the differences 
the differences of the differences, and so on. Similarly, in the har- 
monic triangle, with respect to the elements of any line, those below 
represent the differences, the differences of the differences, and so on; 
those above are the sums, the sums of the sums, and so on — reading 
from the right, however. Thus in these triangles we see that sums and 
differences are the inverses of each other, 64 just as the problem of 
tangents, which depends on the differences of ordinates, is the inverse 
of that of quadratures, which depends on the sum of all the ordinates, 
in the sense of Cavalieri. However, whereas the differences between 
elements in the arithmetic and harmonic triangles are finite, those 
between the ordinates of a curve are infinitesimal, and the formulas 
applicable in the former case no longer hold in the case of curves. 

63 See Mathematische Schriften, V, 108, 404-5; see also Newton, Opera omnia, IV, 512-15, 
and Gerhardt, Die Entdeckung der Differentialrechnung dutch Leibniz, pp. 54-56. 

M See Leibniz, Early Mathematical Manuscripts, p. 142, and Mathematische Schriften, 
V, 397. 



Newton and Leibniz 205 

It was therefore necessary for Leibniz to develop a method of 

procedure for determining sums and differences of infinitesimals. This 

he appears to have done by about 1676, the time at which Newton 

composed De quadrate ra. He had, about a year before, adopted his 

characteristic notation. He employed $x, or later fxdx, for the "sum" 

of all the x's — or the "integral" of x, as he called it later, on the 

suggestion of the Bernoulli brothers. 55 For the "differences" in the 

x 
values of x, he wrote dx, although he had at first used - for this, in 

d 

order to imply that finding the "difference" involved a lowering of 

the dimension of the quantity. 

Just as in the Principia Newton had begun by finding the moment 

of the product AB, so Leibniz determined the "difference" of the 

product xy. Although at the outset Leibniz was uncertain about his 

method and hesitated as to whether or not d{xy) is the same as dxdy 

( X \ Q/OC 

and whether d[-\ is equal to — , 56 he in the end answered these 
\yj dy 

questions correctly, determining that d(xy) = xdy + ydx and dl — \ 

'vdoc "—* ocd/v 
= -. These values he found by allowing x and y to become 

y 2 

x + dx and y + dy respectively. Upon subtracting the original value of 
the function from the new one and observing that dxdy is infinitely small 
in comparison with the terms xdy and ydx, the results are obtained. 

Having established the rules for differences and quotients, Leibniz 
was then able to extend these to all integral powers of a variable, the 
difference of x n being nx n ~ l dx. Because summation is the inverse of 

determining a difference, the integral of x is, of course, ." 

n + 1 

We have had occasion to observe the derivation of the latter result in 

various forms in the methods of previous investigators, but only in 

the work of Newton and Leibniz was it obtained as the inverse of 

another fundamental operation. A comparison of the derivations by 

M See James Bernoulli, "Analysis problematis antehac propositi," p. 218. 

66 Leibniz, Early Mathematical Manuscripts, p. 102; Gerhardt, Die Entdeckung der 
Differentialrechnung durch Leibniz, pp. 24, 38. 

67 Cf. Mathematische Schriften, V, 226 ff. 



206 Newton and Leibniz 

the methods of fluxions and differences with those by quadratures as 
given by Cavalieri, Torricelli, Roberval, Pascal, Fermat, and Wallis 
will convince one of the enormous operational facility to be gained by 
such a method of procedure. As Newton made the rules for fluxions 
basic in his method, so also Leibniz looked upon the operation of 
finding "differences" as fundamental in his "differential and sum- 
matory" calculus. Ever since Newton and Leibniz invented their 
methods and combined them in this manner with the discovery of the 
fundamental inverse property, this point of view has continued in 
the elementary calculus. Differentiation is in general the fundamental 
operation, integration being regarded simply as the inverse of this. 

There has been retained also in the calculus a certain element of 
confusion in terminology which is the result of the somewhat different 
attitudes of Newton and Leibniz with respect, not to the determination 
of the integral, but to its definition. Newton defined the fluent as the 
quantity generated by a given fluxion — that is, as the quantity having 
a given magnitude as its fluxion, or as the inverse of the fluxion. Ir 
keeping with this emphasis upon the indefinite integral, Newton 
included in both the Methodus fluxionum and De guadratura what 
amounts to a table of integrals. Leibniz, on the other hand, defined 
the integral as the sum of all the values of a magnitude, 58 or the sum 
of an infinite number of infinitely narrow rectangles, or — as modern 
mathematics would express it — as the limit of a certain characteristic 
sum. These two points of view have been perpetuated in the elementary 
calculus, in which there are two integrals: the indefinite integral and 
the definite integral. The origins of these, in the history of the subject, 
are even now sometimes brought vividly to mind in referring to the 
former as "the integral in the sense of Newton" and to the latter as 
"the integral in the sense of Leibniz." 69 Too much stress should not 
be put upon such a distinction, however, because Newton and Leibniz 
were both well aware of the two aspects of the integral. 60 

68 "Seu data differentia dy invenire terminum y, est invenire summas omnium differ- 
entiarum seu dy." See Leibniz, "Isaaci Newtoni tractatus duo." Cf. also Gerhardt, Die 
Entdeckung der Dijjerentialtechnung dutch Leibniz, p. 45. 

69 Cf. Saks, Theotie de VinUgrale, p. 122. 

80 Hoppe ("Zur Geschichte der Infinitesimalrechnung," pp. 186-87), misled by Newton's 
symbol o, has completely misinterpreted the facts on this point. He would have Newton 
make the summation of indivisibles fundamental and have Leibniz oppose this point of 
view by emphasizing the operation of differentiation. The situation is, on the contrary, if 
anything quite the reverse of this. 



Newton and Leibniz 207 

Newton and Leibniz are known as the founders of the calculus largely 
because they established, in the periods 1665-66 and 1673-76 respec- 
tively, the methods and relationships outlined above. There was an- 
other aspect of their work which the inventors felt carried great 
weight — the generality of their methods. Both men pointed out that, 
unlike the anticipatory procedures, their methods were applicable 
even in the case of radicals. The justification for such an assertion 
was made by Newton largely upon the basis of infinite series. If 
(x + o) n is to be expanded by the binomial theorem, the number of 
terms will be infinite for values of n which are not positive integers. 
No conclusion can in general be drawn from an application of the 
theorem in this case unless the series is convergent, but neither Newton 
nor his successors for a century later fully appreciated the need for 
investigations into the question of convergence. Leibniz in this respect 
had perhaps even less caution than many of his contemporaries, for 
he seriously considered whether the infinite series 1 — 1 + 1 — 1 
+ 1 — . . . was equal to ^. G1 He had as well fewer scruples than Wallis 
in the wide generalization of rules demonstrated only for a small 
number of special cases. Although he had indicated a proof of his 
rule for the "difference" of x n for integral values of n only, he an- 
nounced that this would hold for all values, and entitled the first 
printed account of the calculus "A New Method for Maxima and 
Minima, as Well as Tangents, Which Is Not Obstructed by Fractional 
or Irrational Quantities." 62 

This first published treatise of the calculus, a memoir of six pages 
appearing in the Acta erudilorum of 1684, three years earlier than 
Newton's first account, 63 must have repelled most readers seeking an 
introduction to the new method. Even to the Bernoulli brothers, who 
did so much to popularize the subject in its early stages, it was "an 
enigma rather than an explication." 64 In the first place, it contained 
many misprints. 65 Secondly, Leibniz here imitated the bareness and 

61 Cf. Mathematische Schriften, V, 382-87. 

62 "Nova methodus pro maximis et minimis, itemque tangentibus, que nee fractas nee 
irrationales quantitates moratur, et singulare pro illis calculi genus." Acta erudilorum, 
1684, pp. 467 ff. See Mathematische Schriften, V, 220 ff. For an English translation of this 
see Raphson, History of Fluxions, pp. 19-27. 

03 See Matliematische Schriften, V, 220 ff . 

64 See Leibniz, Matliematische Schriften, III (Part 1), 5, n. 

65 Enestrom, "Uber die erste Aufnahme der Leibnizschen Differentialrechnung." 



208 Newton and Leibniz 

simplicity of the Geometrie of Descartes, although in later articles he 
attempted fuller explanations. His 1684 work contained, besides the 
definition of the "difference" or differential of a quantity, the rules — 
without proof — for the differentials of sums, products, quotients, 
powers, and roots, with a few applications to tangents and to prob- 
lems in maxima and minima and points of inflection. It is interesting 
that among these examples is one in which Leibniz derived the law 
of refraction, using Fermat's principle. This suggests the influence of 
Fermat, who had himself given a similar demonstration and whose 
work was repeated later by Huygens; but Leibniz did not refer to 
him here. Although quadratures were not included in this first paper, 
Leibniz developed the applications to such problems two years later, 
in another paper in the Acta eruditorum.™ In subsequent articles in 
this and other journals, 67 Leibniz gave further developments and 
applications of his calculus — such as the determination of the differ- 
entials of logarithms and exponentials, and of osculating figures. 

In all of this work Leibniz realized that he was creating a new 
subject. It has been suggested that it was only after the method of 
Leibniz had achieved marked success that Newton came to regard 
the method of fluxions as constituting a new subject and an organized 
mode of mathematical expression, 68 rather than simply as a useful 
modification of some earlier rules. This is belied by the fact that 
Newton had by 1676 written out three accounts of his method; but it 
remains true, nevertheless, that Leibniz expressed himself more vig- 
orously on this point than did Newton. He said that his analysis was 
to be compared with the methods of Archimedes in much the way 
that the work of Viete and Descartes had been to the geometry of 
Euclid, 69 in that it dispensed with the necessity of imagination. In 
order to popularize it, he announced explicitly all the rules of opera- 
tion, even the simplest, 70 presenting these as though they were rules 
of algebra, and pointing to the reciprocal relation of powers and 
roots as analogous to that subsisting between his "sums" and "dif- 
ferences," or integrals and differentials. 71 

66 "De geometria recondita et analysi indivisibilium atque infinitorum." Acta erudito- 
rum, 1686, pp. 292-300; see also Leibniz, Mathematische Schriften, V, 226 ff. 

67 See Leibniz, Mathematische Schriften, V, for these papers. 

68 De Morgan, Essays on the Life and Work of Newton, pp. 32-34. 

69 Mathematische Schriften, II, 123. 

70 Zeuthen, "Notes,"' 1895, p. 236. 71 Mathematische Schriften, V, 231, 308. 



Newton and Leibniz 209 

This didactic spirit of Leibniz is in contrast with the reticence which 
Newton displayed on the subject of his method of fluxions, perhaps 
from a morbid fear of opposition. With reference to the logical and 
philosophical justifications of his procedures, on the other hand, 
Leibniz was less emphatic than Newton. He did not make a really 
serious effort on this point, because he felt that the calculus, as a 
modus operandi, brought its demonstrations with it. He did not wish 
to make of the infinitely small a mystery, as had Pascal; nor did he 
turn to geometrical intuition for clarification. In appealing only to 
intelligence, he stressed rather the algorithmic nature of the method, 
as he himself spoke of it. In this sense he may justly be considered one 
of the founders of formalism, as opposed to intuitionism, in mathe- 
matics. He had confidence that if he formulated clearly the appro- 
priate rules of operation and that if these were properly applied, some- 
thing reasonable and correct would result, 72 howsoever doubtful might 
be the meaning of the symbols involved. This attitude reflects well the 
corresponding difficulties experienced at the time with imaginary 
numbers. Leibniz, unlike Aristotle, seemed to feel that his position 
was to be justified by an appeal to the principle of sufficient reason to 
determine, in this connection, the transition from possibility to 
actuality. 73 The importunateness of his contemporaries, however, made 
it necessary for him now and then to attempt a further clarification 
of the basic conceptions of his differential calculus. In this respect he 
was neither lucid nor consistent. 74 

From his earliest to his latest work Leibniz made use of the prin- 
ciple that in a relationship containing differentials of various orders, 
those of lowest order only are to be retained, because all the others 
will be infinitely small with respect to these. This is, in a new algebraic 
form, essentially the same as the doctrine which Roberval and Pascal 
had employed when they had held that a line is as nothing in com- 
parison with a square. Leibniz had carried over into analysis the 
idea of infinitely small quantities of different orders, based upon a 
principle of homogeneity which geometrical considerations had sug- 
gested. Whereas Fermat, Barrow, and Newton had made use only of 

72 Klein, Elementary Mathematics from an Advanced Standpoint, p. 215. 

73 Enriques, Historic Development of Logic, p. 77. 

74 Hoppe ("Zur Geschichte der Infinitesimalrechnung," p. 184) has maintained that 
the thought of Leibniz was much deeper and more accurate than that of Newton, but his 
view is based upon a misinterpretation of the work of the latter. 



210 Newton and Leibniz 

first order infinitesimals, Leibniz conceived of an infinite number of 
such orders, corresponding in a sense to the infinite ranks in the 
system of monads found in his philosophical scheme. However, in his 
definition of the differential of the first order, Leibniz vacillated; and 
for those of higher order he was very far from giving a satisfactory 
explanation. 

In the first published account of the calculus, Leibniz gave a 
singularly satisfactory de finiti on of his first-order differentials. He 
said that the differential dx of the abscissa x is an arbitrary quantity, 
and that the differential dy of the ordinate y is defined as the quantity 
which is to dx as the ratio of the ordinate to the subtangent. Barrow's 
rule for tangents had, in a sense, implied a similar definition, for it 
had required the substitution of the ordinate for a and the subtangent 
for e; but Barrow's a and e were vague infinitesimals. In the definition 
of Leibniz given above, the differentials are finite, assignable quantities, 
entirely comparable with those defined in the calculus of today. This 
fact has led to the assertion that "Leibniz from the beginning of the 
new calculus defined the differential absolutely as did Cauchy." 75 

In a sense this is true, but such a statement is quite misleading for 
two reasons. In the first place, the definition of Leibniz presupposes 
logically a satisfactory definition of the tangent line, just as Cauchy's 
differential depends upon the notion of the derivative. In each case it 
would be expected that the explanation should be in terms of limits. 
Leibniz, however, unlike Cauchy, defined the tangent as a line joining 
two infinitely near points of the curve, these infinitely small distances 
being expressible by means of differentials or differences between two 
consecutive values of the variable. 76 This constitutes a petitio principii 
which indicates that the avoidance of infinitely small quantities in the 
thought of Leibniz was only superficial. Of course, it is possible that 
Leibniz intended his language to be interpreted in the precise sense 
of limits, as we do when we speak of the tangent as the line through 
two consecutive or coincident points; but a further consideration of 
Leibniz's writings will make such a view appear a misinterpretation 
of his whole thought. Leibniz, throughout his work, regarded the 

n Mansion, Resume du cours <T analyse infinite" simale, Appendix, "Esquisse historique,' ? 
p. 221; cf. also editorial note, Malhesis, IV (1884), 177. 

78 Mathematiscke Schriften, V, 220 ff.; Gerbardt, Die Entdeckung der Differentialrechnung 
dutch Leibniz, p. 35. 



Newton and Leibniz 211 

differential as fundamental, as a recent study has shown. 77 Modern 
mathematics agrees with Cauchy, however, in making this notion 
subordinate to that of a limit by denning it in terms of the derivative. 
The reason for this change of view is to be found in the failure of 
attempts made by Leibniz and others to give a satisfactory definition 
of the differential independent of the method of limits. 

The attempts of Leibniz to give satisfactory definitions of differ- 
entials of higher orders were unsuccessful. He said that ddx or d 2 x 
is to dx as dx is to x™ making no distinction between the differentials 
of independent and dependent variables. Somewhat similarly he said 
that if dx : x = dh : a, where a is a constant and dh is a constant 
differential, then d 2 x : dx = dh : a or d 2 x : x - dh 2 : a 2 ; and in general 
d e x : x = dti : a e , where e may even be fractional. 79 Perhaps realizing 
that this definition could not be consistently applied, he later gave a 
geometrical interpretation which, although lacking precision of state- 
ment, can be correctly interpreted in terms of derivatives. Given any 
curve, let dx be at every point an assignable quantity and let dy be 
such that the ratio of dy to dx is that of the ordinate to the subtangent. 
If then for every point on the curve we plot on the same axes a new 
point whose ordinate is proportional to dy, the result will be a new 
curve, the differentials of which will be the "differentio-differentials," 
or "second differentials," of the original curve. 80 This geometric rep- 
resentation is in general equivalent to that which would be obtained 

dy 
by plotting the values of the ratio — for every point. The second 

dx 

differential d 2 y would then be determined from the derivative of the 
new curve — that is, from the second derivative of the original curve. 
Leibniz, however, did not regard derivatives as fundamental, so that 
his remarks here cannot be regarded as constituting a satisfactory 
definition of d 2 y, any more than can that of dy in terms of the tangent. 
The lack of such suitable definitions led to bizarre uses of the differ- 
ential symbolism. In 1695 John Bernoulli, in a letter to Leibniz, 

3 / — d 3 y 

wrote such expressions as: V d 6 y = d 2 y and — = d?yd~ 2 x = d 3 ypx. m 

d 2 x 

77 Petronievics, "tJber Leibnizens Methode der direkten Differentiation." 

78 Mathematische Schriften, V, 325; cf. also III (Part 1), 228. 

79 Ibid., Ill (Part 1), 228. » "Addenda ad schediasma proximo," p. 370. 
« Leibniz, Mathematische Schriften, III (Part 1), 180. 



212 Newton and Leibniz 

In the absence of satisfactory definitions, Leibniz resorted fre- 
quently to analogies to clarify the nature of his infinitely small dif- 
ferentials. At one point he made use of the imagery of Newton and 
spoke of his differentials as the momentary increments or decrements 
of quantities. 82 Again he applied the thought of Hobbes and said that 
the conatus is to motion as a point to space or as one to infinity. 83 
The infinitely small he considered the study of the vanishing or incip- 
iency of magnitudes, as opposed to quantities already formed. 84 He 
applied this analogy to second-order differentials as well. If in nature 
a motion is thought of as pictured by a line, then the impetus or 
velocity is represented by an infinitely small line, and the acceleration 
by a line doubly infinitely small. 86 

Appealing again to geometrical intuition, Leibniz said that as a 
point added nothing to a line, because it is not homogeneous or com- 
parable, so differentials of a higher order in his method may likewise 
be neglected. 86 Pursuing this thought further, he said that if one 
thinks of geometrical magnitudes as represented by the ordinary quan- 
tities of algebra, then the first differentials refer to tangents, or the 
directions of lines, and the higher differentials refer to osculations or 
curvatures. 87 In this sense the differentials were to be compared with 
the Euclidean angle of contact which was less than any assignable 
magnitude and yet not zero. 88 

In the absence of rigorous definitions Leibniz continued to multiply 
analogies. In a somewhat less critical vein he said that the differential 
of a quantity can be thought of as bearing to the quantity itself a 
relationship analogous to that of a point to the earth or of the radius 
of the earth to that of the heavens. 89 In another place he said that 
as the earth is infinite with respect to a ball held in the hand, so the 
distance of the fixed stars is doubly infinite with respect to the ball; 90 
and this analogy he repeated later, substituting a grain of sand for the 
ball. 

John Bernoulli, the pupil of Leibniz, with great naivet6 pointed 
out other analogies which the work of Galileo and of Leeuwenhoek 

M Mathematische Sckriften, VH, 222. 

M Pkilosophische Sckriften, IV, 229. Cf . also Mathematische Schrijten, III, 536 ff. 

M Pkilosophische Sckriften, VI, 90. " "Testamen de motuum coelestium," p. 86. 

» Mathematische Sckriften, V, 322. * Ibid., V, 325-26; cf. also p. 408. 

88 Mathematische Sckriften, V, 388. a "Testamen de motuum coelestium," p. 85. 

*> Mathematische Sckriften, V, 350, 389. 



Newton and Leibniz 213 

had made possible. He compared the orders of the infinitely great with 
the relations of the stars to the sun, to the planets, to the satellites of 
these, to the mountains on the last named, and so on. The infinitely 
small quantities in the same way resembled the apparently number- 
less grades of animalcules which the microscope had disclosed. 91 
Leibniz, while appreciating the comparison, cautioned Bernoulli that 
the latter were of finite size, whereas the differentials were infinites- 
imals. 92 

It is interesting to notice that whereas Newton had used the 
infinitesimal conception in his early work, only to disavow it unequiv- 
ocally later and to attempt to establish the idea of fluxions on the 
doctrine of prime and ultimate ratios of finite differences — that is, 
in terms of limits — we shall find that with Leibniz the tendency is 
somewhat in the other direction. Beginning with finite differences, he 
was to be confirmed in his use of infinitesimal conceptions by the 
operational success with which his differential method met, although 
he seems to have remained largely in doubt as to its logical justifica- 
tion. The divergent views of these two men were perhaps less the result 
of dissimilar mathematical traditions than of varying tastes. Newton, 
the scientist, found in the notion of velocity a basis which to him 
appeared to be satisfactory; Leibniz, the philosopher, who was, as 
well, perhaps as much a theologian as a scientist, 93 preferred to find 
this in the differential, the counterpart in thought of the monad, 
which was to play such a large part in his metaphysical system. 

Although Leibniz continued to employ the conceptions and the 
methods of infinitesimals, his justification of the infinitely small was 
not at first the result of serious effort, inasmuch as the notion had 
been employed by geometers with more or less indulgence throughout 
the century. Although Huygens and some few others did not readily 
accept the new calculus, they had not opposed it. In 1694, however, 
the Dutch physician and geometer, Bernard Nieuwentijdt, who was 
in this respect to be succeeded by a long line of mathematicians, 
opened an attack upon the lack of clarity in Newton's work and 
upon the validity of the higher differentials of Leibniz. Although he 
admitted in general the correctness of the results of the new methods, 

91 Leibniz and Bernoulli, Cotnmercium philosophicum et ntathematicum, I, 410 ff. 
<* Ibid.; cf. also Mathematische SckrifUn, HI (Part 2), 518, 524. 
93 Mach, Science of Mechanics, p. 449. 



214 Newton and Leibniz 

he felt that they entailed some obscurity and said that they not infre- 
quently led to absurdities. The tangent procedure of Barrow he 
criticized because a and e were taken as zero. 94 Newton's evanescent 
quantities he regarded as too vague, and he said that he could not 
follow the reasoning in the lemmas on limits, by which quantities, 
which in a given time tend to equality, are in the end equal. 95 In the 
analysis of Leibniz he questioned the manner in which the sum of 
infinitesimals might be a finite quantity. 96 

In a further attack in the following year Nieuwentijdt said that 
Leibniz had not made the nature of infinitely small quantities any 
clearer than had Newton and Barrow, nor could he explain in what 
way the differentials of higher order differed from those of the first 
order. 97 Nieuwentijdt tried to develop a method which would solve 
the problems of Leibniz without using infinitesimals of higher order; 
but the result was a failure. 98 The infinite and infinitesimal Nieu- 
wentijdt defined unsatisfactorily, as had Nicholas of Cusa, as quan- 
tities respectively greater than and less than any given magnitude. 99 
Thus, whereas the work of Newton and Leibniz had tended, in certain 
respects, toward the limit concept, the view of Nieuwentijdt repre- 
sented a reversion to a less critical manipulation of the static infinite 
and the infinitesimal, which in the following century was boldly 
extended to the infinitely large and the infinitely small of higher 
orders, particularly by Fontenelle. This was unfortunate, as a serious 
effort to establish a secure foundation for the calculus was to be desired 
at the time. 

Leibniz answered Nieuwentijdt in 1695 in the Acta eruditorum, 100 
defending himself from "overprecise" critics, whom he likened to the 
Skeptics of long before. He maintained that we should not be led by 
excessive scrupulousness to reject the fruits of invention. His method, 
he maintained, differed from that of Archimedes only in the ex- 
pressions used, being in this respect more direct and better adapted 
to the art of discovery. 101 After all, he held, the phrases "infinite'' 

94 Nieuwentijdt, Consideraiiones circa analyseos ad quantitates infinite parvas applicatae 
f/rincipia, p. 8 

98 Ibid., pp. 9-15. »« Ibid., p. 34; cf. also pp. 15-24. 

97 Analysis infinitorum seu curvilineorutn proprietates ex polygonorum natura deduclae. 

98 See Weissenborn, Die Principien der hoheren Analysis, Section 10. 

99 Analysis infinitorum, p. 1. 

190 See Mathematische Sckriften, V, 318 ff. m Mathematische Sckriften, V, 350. 



Newton and Leibniz 215 

and "infinitesimal" "signify nothing but quantities which one can 
take as great or as small as one wishes" — somewhat as Aristotle had 
regarded the infinitely small as indicating a potentiality only — "in 
order to show that an error is less than any which can be assigned — 
that is, that there is no error." 102 Nevertheless, he reiterated that one 
can use these as "ultimate things" — that is, as actually infinite and 
infinitely small quantities — as a tool, much "as algebraists retain 
imaginary roots with great profit." 103 This Janus-faced appearance of 
the differential recurs frequently in the work of Leibniz. Although he 
commonly spoke of it as infinitely small, he often used instead the term 
incomparably small. 104 He held that "if one preferred to reject infi- 
nitely small quantities, it was possible instead to assume them to be 
as small as one judges necessary in order that they should be incom- 
parable and that the error produced should be of no consequence, or 
less than any given magnitude." 105 

At another point Leibniz said that the differential was "less than 
any given quantity" and compared the neglect of differentials to the 
work of Archimedes, who "assumed, together with all those following 
him, that quantities which did not differ by a given quantity were in 
fact equal." 106 He here seemed to feel that his method constituted but 
a cryptic form of expression of the method of exhaustion, so that 
further justification was superfluous. However, the meaning of the 
symbols themselves was still not clear. 

The hesitation which Leibniz displayed as to whether the differ- 
entials were to be regarded as assignables or inassignables is perhaps 
most neatly illustrated by a procedure he suggested in his Historia et 
origo calculi differentialis, written a year or two before his death. In 
order to avoid working with quantities which are not really infini- 
tesimals but which are treated as such — amphibia between existence 
and nonexistence, as Leibniz called the imaginary numbers — he re- 
sorted to a bit of sleight of hand. Leibniz here used the symbols 
{d)x and (d)y to represent finite assignable differences; then, after the 
calculation had been completed, he replaced them by the inassignable 

m PhUosophische Schriften, VI, 90. 

108 Leibniz, Early Mathematical Manuscripts, p. 150. 

104 Mathematische Schriften, V, 407; cf. also p. 322. 

106 "Testamen de motuum coelestium > ,, p. 85. 

"* "Responsio ad nonnullas difficultates, a Dn. Bernardo Nieuwentijt," p. 311. 



216 Newton and Leibniz 

infinitesimals or differentials dx and dy, "as a kind of fiction," because, 
after all, "dy : dx can always be reduced to a ratio (d)y : d(x) between 
quantities which are without any doubt real and assignable." 107 

Just how this jump from assignables to inassignables and back 
again is to be justified, Leibniz did not make plain. It appears from 
this argument that he realized that it was not the individual differ- 
entials, but only their ratio, that constituted the important considera- 
tion. Newton likewise understood that the significance of his method 
lay in the ratio of fluxions, so that for fluxions one could substitute 
other finite quantities in the same proportion. That the limit concept 
does not stand out clearly in their work is probably the result of the 
fact that they, and their contemporaries, were thinking always of a 
ratio as the quotient of two numbers, rather than as a single number 
in its own right. 108 Only after the development of the general abstract 
concept of real number was the way clear to interpret both the 
fluxionary and the differential calculus in terms of the limit of an 
infinite sequence of ratios or numbers; but this interpretation did not 
become generally acceptable for another century. The meanings of 
the terms "evanescent quantities" and "prime and ultimate ratio" 
had not been clearly explained by Newton, his answers being equiv- 
alent to tautologies: 

But the answer is easy: for by the ultimate velocity is meant that, with 
which the body is moved, neither before it arrives at its last place, when 
the motion ceases, nor after; but at the very instant when it arrives. . . . 
And, in like manner, by the ultimate ratio of evanescent quantities is to be 
understood the ratio of the quantities, not before they vanish, nor after, 
but that with which they vanish. 109 

This sounds very much like the quotient of two infinitesimals, al- 
though Newton a,dded that "those ultimate ratios with which such 
quantities vanish are not truly the ratios of ultimate quantities but 
the limits to which the ratios of quantities, decreasing without end, 
always converge." In other words, it is the ratio in which Newton 
was interested, not the evanescent quantities themselves; but he failed 
to define this ratio unequivocally. 

107 See Leibniz, Early Mathematical Manuscripts, p. 155; Weissenborn, Die Principien 
der hoheren Analysis, p. 104; Gerhardt, Die Entdeckung der Differentialrechnung durch 
Leibniz, p. 31. 

108 Pringsheim, "Nombres irrationnels et notion de limite," pp. 143-44. 
108 Opera omnia, I, 250-51; II, 40-41. 



Newton and Leibniz 217 

Leibniz had a somewhat similar idea. Just as Newton never cal- 
culated a single fluxion but always a ratio, so also Leibniz realized 
that it was the ratio of, or relationship between, the differentials 
which was significant. These could therefore be regarded as any finite 
quantities, the ratio of which was that of the ordinate to the sub- 
tangent. For pragmatic reasons, however, Leibniz retained the infi- 
nitely small, justifying this by saying that if one desired rigor, he could 
for the "inassignables" substitute "assignables" having the same 
ratio. However, just as Newton did not clearly explain how ratios of 
evanescent quantities become, or are related to, "prime" or "ultimate" 
ratios, so Leibniz was unable to explain the transition from finite to 
infinitesimal magnitudes. Leibniz admitted that one could not prove 
or disprove the existence of infinitely small quantities. 110 

Furthermore, Leibniz felt that the justification for his calculus lay 
in the ordinary mathematical considerations already known and used, 
and that it was "not necessary to fall back upon metaphysical contro- 
versies such as the composition of the continuum." 111 Nevertheless, 
when called upon to explain the transition from finite to infinitesimal 
magnitudes, he resorted to a quasi-philosophical principle known as 
the law of continuity. We have seen previous applications made of 
this doctrine by Kepler and by Nicholas of Cusa. The latter may have 
influenced Leibniz in this respect, as well as in the philosophical 
doctrine of monads. 112 

Leibniz, however, gave to the doctrine of continuity a clarity of 
formulation which had previously been lacking and perhaps for this 
reason looked upon it as his own discovery. This "postulate" Leibniz 
expressed, in a letter to Bayle in 1687, as follows: "In any supposed 
transition, ending in any terminus, it is permissible to institute a 
general reasoning, in which the final terminus may also be included." 113 
Thus in his manipulations in the calculus, "the difference is not 
assumed to be zero until the calculation is purged as far as is possible 
by legitimate omissions, and reduced to ratios of nonevanescent 
quantities, and we finally come to the point where we apply our result 

m Leibniz and Bernoulli, Commercium philosophicum et tnathematicum, I, 402 ff.; cf. also 
Mathematische Schriften, III (Part 2), 524 ff. 

m Leibniz, Early Mathematical Manuscripts, pp. 149-50. 

m See Zimmermann, "Der Cardinal Nicolaus Cusanus als Vorlaufer Leibnizens." 

m Leibniz, Early Mathematical Manuscripts, p. 147; see also Philosophische Schriften, 
TJl y 52; Mathematische Schriften, V, 385. 



218 Newton and Leibniz 

to the ultimate case" 114 — ostensibly by virtue of the law of continuity. 
Thus even in the work of Leibniz the idea of a limit was implicitly 
invoked, although the logical order was reversed. Leibniz justified the 
limiting condition by the law of continuity, whereas mathematics has 
since shown that the latter must itself first be defined in terms of 
limits. In this manner of thinking Leibniz seems still to be striving to 
make use of a vague idea of continuity which we feel we possess and 
which had bothered thinkers since the Greek period. 

Newton, of course, had tacitly ensconced part of his difficulty with 
the continuum in the comforting notion of continuous motion, al- 
though he too, in his prime and ultimate ratio, was implicitly invoking 
the law of continuity of Leibniz. While Newton sought to avoid the 
limit concept by notions acceptable to scientific empiricism, Leibniz 
had recourse to ideas of ultimate form, suggested by metaphysical 
idealism. Even when the quantities — such as differentials — involved in 
a relationship become inassignable, nevertheless he felt that the ulti- 
mate form remained. A point, he held, was not that whose part is 
zero, but whose extension is zero. 115 In a similar connection Leibniz 
had asked Wallis, "Who does not admit a figure without magnitude?" 116 
The characteristic triangle was for Leibniz one in which the form of 
the triangle remained after all magnitude had been abstracted from 
it, 117 much as Newton had spoken of the ultimate forms of evanescent 
triangles. 118 

Leibniz had intended to write a volume on the subject of the 
infinite. This book, which would have constituted a definitive ex- 
pression of his views, did not appear. However, the attitude which 
Leibniz expressed in connection with his law of continuity seems to 
have been that which, with some vacillations and modifications, he 
held until his death. In his Theodicee he had said, in speaking of 
infinite and infinitely small quantities, "but all that is nothing but 
fictions; every number is finite and assignable, as is every line like- 
wise." 119 Nevertheless, several years later, in writing to Grandi, he 
said, "Meanwhile, we conceive the infinitely small not as a simple 

n * Leibniz, Early Mathematical Manuscripts, pp. 151-52. 

U5 PkUosophische Schriften, IV, 229. 

u « Maihematische Schriften, IV, 54; cf. also IV, 63. 

w Cf. Freyer, Studien zur Metaphysik, p. 10. 

m Opera omnia, I, 243. 1W Philosophische Schriften, VI, 90. 



Newton and Leibniz 219 

and absolute zero, but as a relative zero, (as you yourself well remark), 
that is, as an evanescent quantity which yet retains the character of 
that which is disappearing." 120 

Leibniz here clearly has in mind the law of continuity. This prin- 
ciple, the origin of which lay in the nature of the infinite, he felt to be 
absolutely necessary in geometry, as well as useful in physics. 121 
Accordingly, he regarded an equality as a particular case of an 
inequality, and an infinitely small inequality as becoming an equality. 122 
Acceptance of the law of continuity would therefore justify the 
omission of differentials of higher order, and it appears that it is upon 
this basis that Leibniz would have his calculus justified. A rather 
inexact tradition would impute to Leibniz a belief in actually infin- 
itesimal magnitudes. 123 However, Leibniz himself, in a letter written 
about two months before his death, said emphatically that he "did not 
believe at all that there are magnitudes truly infinite or truly infinites- 
imal." 124 These conceptions he regarded as "fictions useful to abbre- 
viate and to speak universally." 125 The link between these fictions and 
reality he undoubtedly felt would be found in his law of continuity, 
which he had taken as basic in all his later work in the calculus. 126 

We have traced the development of the calculus through the 
invention of the methods of Newton and Leibniz, and have found that 
the concepts involved had not yet been clarified. Newton gave three 
interpretations of his procedure, and although indicating a preference 
for the notion of prime and ultimate ratios as the most rigorous, he 
did not elaborate any one into a careful logical system. Leibniz dis- 
played a similar lack of decision, for although he employed the 
infinitesimal method throughout, he wavered in his attitude toward 
differentials, considering them variously as inassignables, as quali- 
tative zeros, and as auxiliary variables. 

It has very misleadingly been said that Leibniz developed his method 
logically, while Newton was changeable in his conceptions, using 
limiting ratios to mask his infinitesimals. 127 The converse can, with 

120 Malhematische Schriften, IV, 218 f. 

121 Philosophised Schriften, III, 52. m Ibid., II, 105. 
m Cf. Klein, Elementary Mathematics from an Advanced Standpoint, p. 214. 

™ Opera omnia (Dutens), HI, 500. ™Ibid. 

m Scholtz, Die exakte Grundlegung der Infinitesimalrechnung bei Leibniz, p. 39. 
i* 7 Hoppe, "Zur Geschichte der Infinitesimalrechnung," p. 184; Moritz Cantor, "Origines 
du calcul infinitesimal," p. 24. 



220 Newton and Leibniz 

perhaps more justice, be asserted. The position of Leibniz was more 
difficult to maintain than had been that of the users of indivisibles, 
for he was required to explain infinitesimals not only of first order, but 
of all orders. Newton had avoided a multiplicity of infinitesimals 
through the method of fluxions, which required only a single incre- 
ment — that in the time, o. His method was consequently well adapted 
to an interpretation in terms of limits, assuming that the idea of 
fluxion has first been properly defined or is recognized as a primary, 
undefined notion. Leibniz, on the other hand, had no quantity to 
serve as an independent variable, so that although he recognized that 
it was not the single differentials but only their ratio which was 
significant, he could not indicate how this fact was to be applied in his 
work. Newton had experienced this same difficulty when, in his Principia 
demonstration, he abandoned the fluxionary method and resorted to a 
use of infinitesimals in connection with a function of two variables. 

Furthermore, the notation of Leibniz concealed, perhaps more 
effectively than that of Newton, the logical basis of the calculus. 
Leibniz had developed — as the result of painstaking investigation, 
patient experimentation, and frequent correspondence on the subject 
with other mathematicians 128 (particularly the Bernoulli brothers) — 
a symbolism which was remarkably felicitous when applied to the 
solution of problems. Because it is so convenient as to be almost 
automatic, this notation has been maintained to the present day. 
Nevertheless, this very success operated to mislead Leibniz as to the 
rigorous formulation of the subject. His system of notation caused him 
to think of his differential ratios as quotients, and his integrals as 
sums — of what, he could not say — rather than as limiting values of 
certain characteristic functions. His view seems to have been that 
first-order differentials were incomparables (a useful fiction), which 
by the law of continuity had the same ratio as that of the ordinate 
to the subtangent. Second-order differentials were the inassignables 
which were to those of first order as the latter were to unity. Such a 

j /- • • d 2 y (dy\ 2 

definition of course confused — with I — 1 , and this unsatisfactory 

0-x \CLdC I 

state of affairs was not cleared up until the infinitesimal calculus was 
explained later in terms of limits. 
128 Cajori, A History of Mathematical Notations, II, 180-81. 



Newton and Leibniz 221 

Newton's notation likewise offered difficulties in the way of logical 
formulation. Not only did the infinitely small insinuate itself into the 
method of fluxions in the form of a "moment," but Newton occasion- 
ally failed to exercise care in distinguishing clearly between fluxions 
and moments. Furthermore, in the terminology of "prime and ultimate 
ratios" there was the idea, which was to persist for a century, that 
the limiting quantities were ratios, rather than single numerical values. 
Newton recognized that the ultimate ratio was a ratio of fluxions and 
not of moments. However, he was not clear in his explanations on 
this point and failed to point out the analogy between this ratio and 
the so-called sum of an infinite series. As the Euclidean view of 
number had been based upon the ratio of geometrical magnitudes, 
so here Newton emphasized the ratio of two velocities, rather than the 
single limiting value of a function, as brought out in the nineteenth- 
century definitions of the derivative. 

With such indecision and lack of clarity on the part of the inventors 
of the calculus, it is hardly surprising to discover confusion among 
their followers as to the nature of the subject. This was intensified 
by the fact that many mathematicians failed completely to distinguish 
between the two systems and so misunderstood the arguments of 
their authors. This lack of distinction was due to some extent to 
Newton and Leibniz themselves. In the Acta eruditorum of 1705 129 an 
account of Newton's De guadratura was given, probably by Leibniz, 
in which it was stated that in this book the differentials of Leibniz 
were merely replaced by fluxions, and it is implied that these are 
essentially the same. In England, the committee of the Royal Society 
investigating the priority claims of Newton and Leibniz reported in 
the Commercium epistolicum of 17 12 130 that "The differential metliod 
is one and the same with the method of fluxions, excepting the name 
and mode of notation; Mr. Leibniz calling those quantities differences 
which Mr. Newton calls moments or fluxions, and marking them with 
the letter d, a mark not used by Mr. Newton." 131 Thus even Newton's 
own countrymen did not realize that although he had used infinitely 

m See Leibniz, "Isaaci Newtoni tractatus duo," p. 34; cf. also Bertrand, "De l'invention 
du calcul infinitesimal." 

1,0 See Newton, Opera omnia, IV, 497-592, for a reprint of this. See also the review 
(probably by Newton), Philosophical Transactions, XXIX (1714-16), pp. 173-224. 

m See Opera omnia, IV, 588-89. 



222 Newton and Leibniz 

small moments in his early work, he professed to have nothing to do 
with them in his later expositions in terms of fluxions and of prime 
and ultimate ratios. This is more easily understood when we recognize 
that Newton himself never admitted a change in his view, and that 
his prime and ultimate ratio, unless interpreted under the limit 
concept or the principle of continuity, will involve the infinitely small. 

The first published volume devoted expressly to the history of the 
calculus — The History of Fluxions by Joseph Raphson — appeared in 
1715 and illustrated well the prevailing confusion of thought. To the 
author of this volume and to Halley, Newton had in 1691 planned to 
entrust the preparation for publication of his De guadratura. m How- 
ever, a change of heart led Newton to withhold publication until 
1704, by which time he had, as has been noted, emphatically re- 
nounced infinitesimals. Raphson, nevertheless, failed to recognize any 
change in the point of view and continued to confuse fluxions and 
moments. The small letters, which in the Principia Newton had 
employed to designate moments, Raphson interpreted as symbols of 
fluxions, and these he identified (all too uncritically) with the infin- 
itesimals of Barrow and Nieuwentidjt, and with the differentials of 
Leibniz. 133 Raphson went so far as to regard fluents as infinitely great 
with respect to the finite quantities from which they were obtained; 
and fluents of higher orders were in similar manner interpreted as of 
higher degrees of infinity. 134 

In all fairness it is to be remarked that the object of Raphson's 
work was not to furnish a clarification of the basic notions of the 
calculus. The book contains, in fact, no formal definitions for such 
terms as fluxion, moment, or fluent. The purpose was partly to present 
the various methods in such a way as to make as easy as possible 
the application of the rules of procedure to specific problems, but 
"chiefly to do justice to their authors in point of chronology." 135 In 
both respects Raphson vigorously maintained the superiority of the 
work of Newton. With respect to priority his views were essentially 
those expressed in the report in the Commercium epistolicum of three 
years earlier, suggesting the possibility of plagiarism on the part of 
Leibniz. 136 

m See Raphson, The History of Fluxions, pp. 2-3. 133 Ibid., p. 4. 

"* Ibid., p. 5. «« Ibid. «« Cf. Ibid., pp. 8, 19, 61, 92. 



Newton and Leibniz 223 

As to the Leibnizian method and notation, Raphson most unjustly 
characterized this as "less apt and more laborious" and as a "far- 
fetched symbolizing" of "insignificant novelties." 137 This unfortunate 
attitude, which excessive deference to the great reputation of Newton 
had engendered, was to prevail among British mathematicians for 
just about a century following the appearance of Raphson's work, 
with the result that the differential notation made little headway in 
England before 1816. This period was characterized likewise by the 
same confusion of fluxions and the infinitely small as found in the 
exposition given by Raphson. As it was, a large number of the textbooks 
on the method of fluxions regularly interpreted fluxions as infinitely 
small quantities, 138 thus adding to the prevailing confusion in the 
interpretation of the conceptual bases of the calculus, which furnished 
the provocation for a century of criticism and controversy anent the 
subject. 

™ Ibid., p. 19. 

138 Cajori, A History of the Conceptions of Limits and Fhixions, Chap. II; De Morgan, 
"On the Early History of Infinitesimals in England." 



VI. The Period of Indecision 

THE FOUNDERS of the calculus had clearly stated the rules of 
operation which were to be observed, and the astonishing success 
of these when applied to mathematical and scientific problems by 
Euler, Lagrange, Laplace, and a host of others led men to overlook 
somewhat the highly unsatisfactory state of the logic and philosophy 
of the subject. Throughout the whole of the eighteenth century there 
was general doubt as to the nature of the foundations of the methods 
of fluxions and the differential calculus. In England Newton's lack of 
clarity and his inconsistency in notation was followed by a confusion 
of fluxions with moments. On the Continent the metaphysical ration- 
alism of Leibniz was neglected by his followers, who freely attempted 
to interpret the differentials as actual infinitesimals or even as zeros, 
and who criticized Leibniz for his hesitancy in this respect. 

Such a state of affairs could not long continue unchallenged. 
Nieuwentijdt had earlier questioned the validity of differentials of 
higher orders. Gassendi had referred to the vanity of mathematical 
demonstrations based upon the infinitesimal methods, 1 and Bayle had 
used the difficulties of the infinite in the service of a general skepti- 
cism. 2 However, the most general and significant attack upon the struc- 
ture of the new analysis was launched in 1734 by the philosopher and 
divine, George Berkeley, in a tract called The Analyst. 3 

Berkeley had previously attacked Newton's cosmology in his 
Essay towards a New Theory of Vision, but the motive prompting his 
animadversions in The Analyst was as largely that of supplying an 
apology for theology as it was of inflicting upon the proponents of the 
new calculus a rebuke for the weak foundations of the subject. This 
we gather from the subtitle of the tract: Or a Discourse Addressed to 
an Infidel Mathematician. [This referred to Newton's friend, Edmund 
Halley.] Wherein It Is Examined Whether the Object, Principles, and 
Inferences of the Modern Anaylsis Are More Distinctly Conceived, or 

1 See Bayle, Dictionnaire kistorique et critique, XV, 63. 

2 Cohn, Geschichte des Unendlichkeits problems, pp. 193-97. 

3 The Works of George Berkeley, Vol. III. 



The Period of Indecision 225 

More Evidently Deduced, than Religious Mysteries and Points of Faith. 
"First Cast the Beam Out of Thine Own Eye; and Then Shalt Thou See 
Clearly to Cast Out the Mote Out of Thy Brother's Eye." 

Berkeley in this work did not deny the utility of the new devices 
nor the validity of the results obtained. He merely asserted, with some 
show of justice, that mathematicians had given no legitimate argu- 
ment for their procedure, having used inductive instead of deductive 
reasoning. 4 His objections were not to the mathematician as an artist 
and computist, but as "a man of science and demonstration" and 
"in so far forth as he reasons." After giving an account of Newton's 
method of fluxions which is eminently fair, except for the irrelevant 
ridicule of successive fluxions, Berkeley pointed out his specific 
objections. Since Newton had admitted to no change in his point of 
view, Berkeley legitimately took advantage of this fact to criticize a 
demonstration in the Principia in which the author had made use of 
infinitely small quantities in determining the moment of a product. 5 
Newton had found the moment of the rectangle AB by giving to 
A and B the decrements and increments \a and \b, then subtracting 
the diminished rectangle from the augmented. Berkeley objected that 
we must take the whole amounts a and b as increments or decrements 
to find the moment of AB, and in this case we would have to omit 
from the final calculation the infinitely small quantity ab. Berkeley 
with perfect right criticized leaving out the infinitely small quantity 
ab, if the result is to be rigorously correct, and cited a passage in Be 
quadratura, in which Newton professed not to neglect anything, no 
matter how small. 

Newton had not gone into the justification of his demonstration, 
but had indicated that one could find this in the method of prime and 
ultimate ratios. Berkeley therefore attacked this type of reasoning, as 
applied by Newton in one of his leading demonstrations: that of finding 
the fluxion of x n in De quadratura, in which Newton had sought to 
avoid the infinitely small. 6 Here it will be recalled that Newton had 
given x an increment o, had expanded (x + o) n by the binomial the- 
orem, had subtracted x n to obtain the increment in x n , had divided 
by o to find the ratio of the increments of x n and of x, and had then 

* Cf. ibid., p. 30. 

5 Ibid., pp. 22-23. 6 Ibid., Ill, 24-25. 



226 The Period of Indecision 

let o become evanescent, thus determining the ultimate ratio of the 
increments (or of the fluxions). Berkeley averred that Newton here 
disregarded the law of contradiction, assuming first that x has an 
increment and then, in order to reach the result, allowing the incre- 
ment to be zero, i. e., assuming that there was no increment. Berkeley 
maintained that the supposition that the increments vanish destroys 
the supposition that they were increments. Such an interpretation of 
Newton's meaning, which of course results in the consideration of the 

indeterminate ratio -, is not unjustified, inasmuch as Newton did 

not sufficiently explain the terms "evanescent quantity" and "prime 
and ultimate ratio," upon which the reasoning depends. The modern 
interpretation in terms of limits, of course, considers the infinite 
sequence formed by the ratios as the increment approaches zero, and 
this has no last term, although it is defined as having a limit. Newton's 
expression "ultimate ratio" is misleading, to say the least, and in all 
events indicates a lack of appreciation of the subtle difficulties involved 
in the concepts of infinity, continuity, and real number — difficulties 
not resolved until the second half* of the nineteenth century. 

Although Berkeley's arguments were directed chiefly against the 
British method of fluxions, the method of differentials, as used on the 
Continent by L'Hospital and others, also came in for criticism. 7 He 
explained that in finding the tangent by means of differentials, one 
first assumes increments; but these determine the secant, not the 
tangent. One undoes this error, however, by neglecting higher differ- 
entials, and thus "by virtue of a twofold mistake you arrive, though 
not at science, yet at the truth." This interpretation of the validity of 
the results of the calculus, as due to a compensation or errors, 8 we shall 
find advanced again by Euler, Lagrange, and Carnot, proponents of 
the differential method who wished to clarify its bases. 

Berkeley's criticism of Newton's propositions was well taken from 
a mathematical point of view, and his objection to Newton's infini- 
tesimal conceptions as self-contradictory was quite pertinent. On the 
other hand, his objections to some of Newton's quantities — such as 
fluxions, nascent and evanescent augments, moments as increments 
in statu nascenti, prime and ultimate ratios, infinitesimals, ultimate 

1 1bid., Ill, 20, 29. « Ibid., Ill, 32. 



The Period of Indecision 227 

forms of evanescent triangles — on the grounds of their putative in- 
comprehensibility or inconceivability, although just as much called 
for, were really misdirected. Assuming that the symbols involved have 
been clearly and logically defined (except for the primary undefined 
elements of the subject), it is of no mathematical consequence whether 
one can conceive of them in some manner corresponding to physical 
perception. Thus Berkeley argued that the concept of velocity de- 
pends upon space and time intervals and that it is consequently 
impossible to conceive of an instantaneous velocity, i. e., of a velocity 
in which these intervals are zero. 9 His argument is of course absolutely 
valid as showing that instantaneous velocity has no physical reality, 
but this is no reason why, if properly defined or taken as an undefined 
notion, it should not be admitted as a mathematical abstraction. It is 
interesting to notice that just as the arch-materialist Hobbes, being 
unable to conceive of lines without thickness, denied them to geom- 
etry, so also Berkeley, the extreme idealist, wished to exclude from 
mathematics the "inconceivable" idea of instantaneous velocity. This 
is in keeping with Berkeley's early sensationalism, which led him to 
think of geometry as an applied science dealing with finite magni- 
tudes which are composed of indivisible "minima sensibilia." 10 

In line with this idea, Berkeley suggested that it would be better 
in the calculus to consider increments than velocities. At any rate, he 
warned, increments and velocities should not be confused as Newton 
had upon occasion done. 11 He accepted Cavalieri's indivisibles, but 
insisted that they were finite in number, saying that divisibility to 
infinity is only a fiction, and that infinitely small magnitudes are in- 
conceivable, for they imply the existence of extension without per- 
ception by the mind through the senses. Berkeley was unable to 
appreciate that mathematics was not concerned with a world of "real" 
sense impressions. In much the same manner today some philosophers 
criticize the mathematical conceptions of infinity and the continuum, 
failing to realize that since mathematics deals with relations rather 
than with physical existence, its criterion of truth is inner consistency 
rather than plausibility in the light of sense perception or intuition. 

Although those of Berkeley's arguments which are based upon the 

9 Ibid., Ill, 19. 10 Johnston, The Development of Berkeley's Philosophy, pp. 82-86. 
11 The Works of George Berkeley, III, 46-47. 



228 The Period of Indecision 

inconceivability of the notions involved lose their force in the light 
of the modern view of the nature of mathematics, it is clear that there 
was an obvious need for a logical clarification of many of the terms 
Newton had used. Berkeley's animadversions, although those of a 
nonmathematician, were successful in making this fact appreciated. 
As a result, there appeared within the next seven years some thirty 
pamphlets and articles which attempted to remedy the situation. 
The first appeared in 1734, a pamphlet by James Jurin, Geometry No 
Friend to Infidelity: or, A Defence of Sir Isaac Newton and the British 
Mathematicians, in a Letter Addressed to the Author of the Analyst by 
Philalethes Cantabrigiensis. This defense is weak in the extreme. Jurin 
maintained categorically that fluxions are clear to those versed in 
geometry. With respect to Berkeley's specific criticism on Newton's 
determination of the moment of AB, Jurin gave two answers: the 
moment ab in Berkeley's explanation he held to be as a pin's head 
to the globe of the earth or of the sun or of the orb of the fixed stars; 
the procedure of Newton, however, he defended by asserting that the 
moment is the arithmetic mean of the increment and decrement! In 
answer to Berkeley's objections to the determination of the fluxion of 
x n as given by Newton in De quadratura, Jurin ingenuously said that 
one is not to let the increment in this case be nothing, but to let it 
"become evanescent" or "be upon the point of evanescence," affirming 
that "there is a last proportion of evanescent increments." 12 Jurin's 
response shows that he had no adequate appreciation either of Berk- 
eley's arguments or of the nature of the limit concept. 

Berkeley answered Jurin in 1735, in A Defence of Freethinking in 
Mathematics, 13 and justly asserted that the latter was attempting to 
defend what he did not understand. 14 In this work Berkeley again ap- 
pealed to the divergence in Newton's views — as presented in De analysi, 
the Principia, and De quadratura — to show a lack of clarity in the ideas 
of moments, fluxions, and limits. Jurin's reply in the same year, in 
The Minute Mathematician, was again evasively tautological. He 
explained that "A nascent increment is an increment just beginning 
to exist from nothing, or just beginning to be generated, but not yet 

12 See Geometry No Friend to Infidelity, pp. 35, 52 ff . See also The Minute Mathematician, 
p. 74. 

13 Works, III, 61 ff. " Ibid., Ill, 78. 



The Period of Indecision 229 

arrived at any assignable magnitude how small soever." 15 By Newton's 
ultimate ratio he understood literally "their ratio at the instant they 
vanish." 16 Instead of explaining Newton's lemma on the moment of 
a product in terms of limits, Jurin allowed himself to become involved 
in the tangle Of infinitesimals and was forced to resort to the idea of 
inassignables of Leibniz, saying that the magnitude of a moment is 
nothing fixed or determinate, but is "a quantity perpetually fleeting 
and altering till it vanishes into nothing; in short it is utterly unas- 
signable." 17 

Berkeley, whose Analyst "marks a turning-point in the history of 
mathematical thought in Great Britain," 18 now dropped out of the 
controversy, 19 but the unsatisfactory nature of Jurin's arguments 
was pointed out by Benjamin Robins in A Discourse Concerning the 
Nature and Certainty of Sir Isaac Newton's Methods of Fluxions and 
of Prime and Ultimate Ratios, as well as in articles in current journals. 20 
As indicated in the title of his book, Robins distinguished, not three 
views in Newton's work, but two: that of fluxions and that of prime 
and ultimate ratios. The former he considered to be the more rigor- 
ous, saying that Newton used the latter only to facilitate demonstra- 
tions. 21 He added that the method of fluxions is established without 
recourse to the method of limits. 22 Robins admitted that Newton's 
use of moments in the lemma of the second book of the Principia 
was such as to allow interpretations resembling the language of 
infinitesimals; but he said that Newton thought it sufficient to indi- 
cate once for all that this can be made conformable to the method of 
prime and ultimate ratios, as presented in the lemmas of the first 
book. 23 

Although Jurin had denied the possibility of infinitely small con- 
stants, he had somewhat hazily espoused infinitely small variables, or 

15 See The Minute Mathematician, p. 19. 16 Ibid., p. 30; cf. also p. 56. 

"Ibid., p. 56. 

18 Cajori, A History of the Conceptions of Limits and Fluxions, p. 89. 

19 He had also answered the tract, A Vindication of Sir Isaac Newton's Principles of 
Fluxions Against the Objections Contained in the Analyst, by J. Walton, but neither the 
Vindication — a tautological paraphrase of Newton — nor Berkeley's answer contains any 
significantly new views. See The Works of George Berkeley, III, 107. 

20 See Robins, Mathematical Tracts, Vol. II. 21 Mathematical Tracts, II, 86. 
» See Gibson, "Berkeley's Analyst," pp. 67-69. 

» Cf. Mathematical Tracts, II, 68 ff . 



230 The Period of Indecision 

vanishing quantities. Robins was more emphatic in his disavowal of 
infinitesimals of any kind and said that Newton's statements involving 
moments are to be interpreted in terms of prime and ultimate ratios. 
For example, whereas Jurin had said that Ab + Ba was equal to 
Ab + Ba + ab when a and b banish, Robins said that Ab + Ba was 
as much of the increment of AB as is necessary for expressing the ulti- 
mate ratio. This indicates that Robins realized the more clearly that 
the logical basis is to be found in the method of limits, although he 
was not clear, inasmuch as the product AB involves two independent 
variables, as to exactly how this was to be applied to the case in hand. 
The limit conception of Robins represents a formulation of ideas 
which Valerio and Tacquet had expressed somewhat vaguely a cen- 
tury earlier — ideas to which Robins referred. 24 It indicates, as well, 
the dependence of this notion, in his thought, upon geometric intui- 
tion, for Robins spoke not only of the limits of "ratios of vanishing 
quantities" but also of the limits of the "forms of changing figures," 
giving as an illustration the circle as the limit of the inscribed regular 
polygon, as the number of sides is indefinitely increased. 25 This con- 
fusion of the arithmetical with the geometrical had been responsible 
for much of the vagueness in the work of Newton and of Leibniz, 
and was to persist during the following century. Nevertheless, Robins 
realized more clearly than did Jurin the nature of the limit concept. 
He recognized that the phrase "the ultimate ratio of vanishing quan- 
tities" was a figurative expression, referring, not to a last ratio, but 
to a "fixed quantity which some varying quantity, by a continual 
augmentation or diminution shall perpetually approach, . . . pro- 
vided the varying quantity can be made in its approach to the other 
to differ from it by less than by any quantity how minute soever, 
that can be assigned," 26 . . . "though it can never be made abso- 
lutely equal to it." 27 

Robins realized, as Jurin did not, that the varying quantity need 
not be considered as finally reaching the fixed quantity as its last 
value, although this latter "is considered as the quantity to which the 
varying quantity will at last or ultimately become equal." 28 In the 
controversy between Robins and Jurin, the question as to whether a 

u Ibid., II, 58. «IWd.,II,54. 

26 Mathematical Tracts, II, 49. a Ibid., II, 54. » Ibid. 



The Period of Indecision 231 

variable was to be considered as necessarily reaching its limit played 
a large part. Robins upheld the negative side; Jurin insisted that there 
are variables which reach their limits and vigorously accused his 
opponent of misinterpreting Newton's true meaning. It is difficult to 
judge from Newton's words exactly what he meant. The phrase 
"ultimate ratio" certainly favors Jurin's interpretation, but to avoid 
the logical difficulties inherent in questions of infinitesimals and the 

meaning of -, it was necessary at the time to accept the more logical 

view of Robins that the variable need not attain its limit. 

Robins has recently been criticized 29 for defining his limit in such a 
way that the variable never reaches its limit, on the grounds that 
although this has certain pedagogical advantages, it involves a less 
general conception of limit than Jurin's, and that under this con- 
ception of a limit Achilles could not overtake the tortoise. This 
criticism has been followed by the assertion that one can assume a 
time rate of doubling of the number of sides of a polygon inscribed in 
a circle, such that the circumference (the limit) is reached by the 
polygon (the variable). Such an argument is entirely beside the 
point. Besides confusing the numerical concept of limit with a geo- 
metrical representation, it merely substitutes an infinite time series 
for Zeno's distance series. This it does under the assumption, ap- 
parently, that this is intuitively more impelling, because of our vague 
idea of the relentless flow of time, which in the Achilles is subordinated 
to the static idea of distance. The question as to whether the variable 
S n reaches the limit S is furthermore entirely irrelevant and ambiguous, 
unless we know what we mean by reaching a value and how the terms 
"limit" and "number" are defined independently of the idea of 
reaching. Definitions of number, as given by several later mathe- 
maticians, make the limit of an infinite sequence identical with the 
sequence itself. Under this view, the question as to whether the 
variable reaches its limit is without logical meaning. Thus the infinite 
sequence .9, .99, .999 . . . , is the number one, and the question, 
"Does it ever reach one?" is an attempt to give a metaphysical argu- 
ment which shall satisfy intuition. Robins could hardly have had 

29 See Cajori, A History of the Conceptions of Limits and Fluxions, Chap. IV. 



232 The Period of Indecision 

such a sophisticated view of the matter, but he apparently realized, 
as Jurin does not appear to have done, 30 that any attempt to let a 
variable "reach" a limit would involve one in the discussion as to the 

nature of -. Thus he is hardly to be criticized for his restriction. 


We shall find that the question as to whether a variable reaches its 
limit or not has no significance, in the light of modern definitions. 
At that time, however, it was important as an indication that mathe- 
maticians still felt that the calculus must be interpreted in terms of 
what was intuitively reasonable, rather than of that which was log- 
ically consistent. This is apparently the reason why Robins considered 
the method of fluxions more satisfactory than that of prime and 
ultimate ratios. Everyone assumed he had a clear idea of instantaneous 
motion, although logically this is defined, as Robins failed to realize, 
in terms precisely equivalent to those needed to make the idea of prime 
and ultimate ratio rigorous. 

That the work of Robins was not fully appreciated in England is 
seen in the fact that although Berkeley's Analyst was frequently 
discussed in the flood of texts on fluxions appearing in 1736 and 
1737, the Jurin-Robins controversy was not referred to. 31 Mathema- 
ticians were still not satisfied with the method of limits and, although 
the discussion on infinitesimals may have discouraged the use of the 
idea of moments, it did not banish them. In 1771 the article on fluxions 
in the Encyclopedia Britannica read: "The fluxion of any magnitude 
at any given point is the increment that it would receive in any given 
time, supposing it to increase uniformly from that point; and as the 
measure will be the same, whatever the time be, we are at liberty to 
suppose it less than any assigned time." 32 The old confusion between 
fluxions and moments had not yet ended, and Robins' clarification of 
Newton's method of prime and ultimate ratios was not sufficient to 
establish this as the basis until after continental influences, in the 
early nineteenth century, had brought about a turning point in British 
mathematics. 

In the meantime, however, numerous attempts, some noteworthy 

30 See Jurin, "Considerations upon Some Passages of a Dissertation Concerning the 
Doctrine of Fluxions Published by Mr. Robins," pp. 68 ff. 

11 Cajori, A History of the Conceptions of Limits and Fluxions, p. 179. 
M /Wtf.,p. 240. 



The Period of Indecision 233 

and others insignificant, were made to find new and more satisfactory- 
forms and arguments in which to present Newton's method. By far 
the ablest and most famous of these was made by Colin Maclaurin. 
In his Treatise of Fluxions, in 1742, he aimed, not to alter the con- 
ceptions involved in Newton's fluxions, but to demonstrate the 
validity of his method by the rigorous procedures of the ancients 33 — 
to deduce the new analysis from a few "unexceptional principles." 34 
Maclaurin professed in the preface of this work that the Analyst 
controversy had given occasion to his treatise. Therefore he proceeded 
with extreme circumspection, omitting the notation of fluxions until 
toward the end of the long two-volume treatise. Like Robins, he 
banished the infinitely small as inconceivable and as "being too bold 
a Postulatum for such a Science as Geometry." 35 He did not, however, 
see any objection to introducing into geometry the idea of an instan- 
taneous velocity, for he felt that there can be no difficulty in con- 
ceiving velocity wherever there is motion. 36 In fact, the mathematical 
sciences, Maclaurin said, included velocity and motion, as well as the 
properties of figures. 37 Time, he conceived with Barrow, to flow in a 
uniform course that serves to measure the changes of all things. 38 

Barrow, however, had defined velocity as the power by which a 
certain space may be described in a certain time, 39 somewhat as 
Aristotle had considered motion a manifestation of a potentiality. 
Maclaurin, on the other hand, tried to define instantaneous velocity 
in a manner recalling the attempt of Oresme: "The velocity of a 
variable motion at any given term of time is not to be measured by 
the space that is actually described after that term in a given time, 
but by the space that would have been described if the motion had 
continued uniformly from that term." 40 Maclaurin realized that if an 
instantaneous speed is "susceptible of measuring, it is only in this 
sense." 41 He recognized, that is, that science deals only with actual 
intervals; but he failed to see that as a mathematical notion instan- 
taneous velocity could be defined by extrapolating beyond sense im- 
pressions, through the limit of an average rate of change as the inter- 
vals approach zero. 

Although Maclaurin considered his explanation of fluxions as a 



83 A Treatise of Fluxions, I, 51 ff. 


34 Ibid., I, Preface. 


3S Ibid.,I, iv. 


*<Ibid., I, iii. 


« Ibid., I, 51. 


3 * Ibid., I, S3. 


*/fttf.,I,54. 


40 Ibid., I, 55. 


"Ibid. 



234 The Period of Indecision 

criticism of the method of differentials, his interpretation of fluxions, 
in terms of the intervals which would be generated if the motion were 
to continue uniformly, left the way open to an explanation of Newton's 
fluxionary procedures in terms of finite differences and limits, by which 
the differential calculus of Leibniz was also to be explained. Brook 
Taylor had recognized the importance of this type of exposition and 
had some years previously composed a book on the subject — his 
Methodus incrementorum directa et inversa. 

Whereas Robins and Maclaurin emphasized in Newton's work the 
interpretations in terms of fluxions, Taylor said that Newton had 
founded his method on prime and ultimate ratios. 42 Newton had 
recognized that the limit of the ratio of the moments was the same 
as the ratio of the corresponding fluxions or velocities. Whereas Robins 
and Maclaurin had assumed that everyone had a clear idea of instan- 
taneous velocity, Taylor felt that it was easier to conceive of moments 
and to obtain the ratios of fluxions from these. 43 In this respect, how- 
ever, his work at times approached closely to an unclear manipulation 
of zeros, resembling the later procedures of Euler. Taylor said that the 
relation of fluxions was to be obtained from that involving finite 
differences. In this respect his view resembled that of modern mathe- 
maticians, although Leibniz said at the time that it was "putting the 
cart before the horse." 44 However, Taylor, like Leibniz, was not clear 
with respect to the transition from finite differences to fluxions, for 
he held that to bring this about one simply wrote zero for the "nascent 
increments." Ultimate ratios, he thought, are those in which the 
quantities are already evanescent and are made zero, 45 an attitude 
which was to appear on the Continent in the work of Euler. 

The view of Taylor on instantaneous velocities was shared by 
Thomas Simpson, author of a popular textbook on the method of 
fluxions which appeared first in 1737 and in an enlarged edition in 
1750. Simpson felt that by taking fluxions as mere velocities, the 
imagination is confined to a point, and, without proper care, insen- 
sibly involved in metaphysical difficulties. 46 Maclaurin recognized this 
difficulty also and said that fluxions were measured by the quantities 

42 Taylor, Methodus incrementorum directa et inversa, Preface. **Ibid. 

44 Malhematische Schriften, III (Part 2), 963. 

45 Methodus incrementorum directa et inversa, Preface, and p. 3. 

46 The Doctrine and Application of Fluxions, pp. xxi-xxii. 



The Period of Indecision 235 

they would generate, if they were to continue uniformly. Simpson 
went beyond Maclaurin and identified fluxions of accelerated quan- 
tities with the increments which would be generated in a given portion 
of time if the "generating celerity" were to continue uniformly. 47 
Simpson therefore followed Taylor in reverting to Newton's use of 
moments. He employed v in place of Newton's o, and as Newton had 
in his De analyst omitted terms containing o, so Simpson dropped 
out powers of v when the points coincided, because these powers he 
felt were due to the acceleration of the motion, 48 whereas, according 
to his definition, the generating celerity was to be regarded as con- 
tinuing uniformly. 

The views of Taylor and Simpson are similar to those which Jurin 
expressed in opposing Robins and serve to indicate the continued 
recurrence and widespread use of infinitesimal quantities in England 
during the century following the time of Newton, and the confusion 
of these with fluxions. There was also another element in British 
thought which operated against an early clarification of the bases of 
the calculus. The method of differences of Taylor failed to exert a 
decisive influence not only because of the novelty of his notation and 
his lack of clarity of expression, but also because it was essentially 
arithmetical and so involved a degree of abstraction which seems to 
have been unwelcome in British mathematics at the time. The basis 
of the method of fluxions remained essentially geometrical in its 
conception of quantity, in spite of Newton's work in infinite series; 
and the tenacity with which English mathematicians clung to the 
idea of velocity was probably due as much to a desire for an intuitively 
satisfying conceptual background as to loyalty to their great prede- 
cessor. This is shown particularly in the treatise of Maclaurin, which 
represents the high point in the rigorous interpretation of the calculus 
in terms of geometrical and mechanical notions. This work, however, 
was as little read as it was widely praised, and it consequently had 
probably no more influence than had Robins' treatises, which it 
resembles in its general ideas. 

Much of the Analyst controversy and the confusion in the inter- 
pretation of the limit concept was due to the lack of a clear distinction 
between questions of geometry and those of arithmetic, and to the 

47 Ibid., p. 1; cf. also p. xxii. a Ibid., pp. 3-4. 



236 The Period of Indecision 

absence of the formal idea of a function. These weaknesses in the 
English view are brought out particularly in Newton's determination 
of the moment of a product, which was the object of some of Berkeley's 
most pertinent strictures. Newton had indicated vaguely that this was 
to be made rigorous by the application of the limit method, which is 
essentially arithmetical, but he had interpreted the product geo- 
metrically as the area of a rectangle. As a consequence, there was no 
indication as to whether the product was to be considered a function 
of one or of two variables. Time was, in a sense, made to take the 
place of an independent variable, but this was merely to aid in the 
conceptual representation and not to reduce the problem to one 
expressible in terms of a function of a single independent variable, 
as is necessary for differentiation. 

We shall find that on the Continent there was a growing tendency 
to link the calculus with the formal concept of a function, instead of 
with the intuitional conceptions of geometry. The ideas developed as 
a result of this trend were not significantly influential in changing 
British views until the beginning of the following century, but mean- 
while there were in England occasional abortive attempts to sub- 
stitute arithmetical devices and ideas for those of geometry and 
dynamics. 49 Here and there the idea was expressed that the operations 
of the new analysis should proceed along the line afforded by the ordi- 
nary methods of arithmetic and algebra, and that the introduction of 
the doctrine of motion was unwarranted and unnecessary. However, 
the efforts to establish an algebraic calculus were characterized by 
lack of rigor in exposition. 

One of the best known and least objectionable of the efforts to base 

the calculus only upon principles received in algebra and geometry, 

"without the aid of any foreign ones relating to an imaginary motion 

or incomprehensible infinitesimals," was made in 1758 by John Landen 

in what he called The Residual Analysis. Instead of computing the 

quotient of fluxions, or of differentials, Landen calculated "the value 

of the quotient of one residual divided by another." 50 By a residual 

was to be understood an expression of the form x — x, or x n — x n . 

Landen's method thus was based upon an uncritical manipulation of 

49 See Cajori, A History of the Conceptions of Limits and Fluxions, Chap. IX. 
60 The Residual Analysis, p. v. 



The Period of Indecision 237 

indeterminate forms. Given a function F(x), Landen found "we shall 
frequently have occasion to assign the quotient of F — F divided by 

m 

x — x. bl For example, the fluxion, or residual quotient, of x n he found 
by writing 

v fv\ 2 IV 



x — v - - 1 

= x n 

X — V 



1+^ + 1, !■+•-• + 

x \xl \x/ 



,' + (:)H)* + (f- 

and then taking v = x. b2 

The method of Landen has been characterized as making use of 
the limit of D'Alembert supposed to be attained instead of a terminus 
which can be approached as closely as desired. 53 Such a judgment is 
most charitable. If Landen possessed the limit concept, he certainly 
hid it most effectively under a misleading notation and terminology 
at a time when there was need for a clear and open recognition of its 
fundamental importance. Although we shall find that D'Alembert 
was at the time urging upon continental mathematicians that the 
logical basis of the differential calculus was to be found in the notion 
of limits, there was in England no strong leader to propagate this 
doctrine, unhampered by geometrical and mechanical superfluities. 
When it finally imposed itself upon British mathematics, it was as 
the result of European developments, to which we must now turn. 

While the English mathematicians were so greatly occupied with 
arguments as to the validity of the views involved in the method of 
fluxions, the differential calculus was rapidly gaining in popularity on 
the Continent. The algorithmic essentials of the differential calculus 
of Leibniz had appeared in the Acta eruditorum for 1684, and those 
of the "calculus summatorius," or integral calculus, followed in 1686. 
Inasmuch as Leibniz, unlike Newton, corresponded extensively with 
numerous mathematicians on the subject of the new analysis, seeking 
for the most suitable forms of notation and presentation, there grew 
up a group of enthusiastic admirers of the subject who were soon 
able to make contributions of their own. 

61 Ibid., p. 5. 

62 Ibid., pp. 5-6; see also A Discourse Concerning the Residual Analysis, pp. 5, 41. 

63 See Cajori, A History of the Conceptions of Limits and Fluxions, pp. 238-39. 



238 The Period of Indecision 

In Switzerland, for example, there were John and James Bernoulli, 
the former of whom in 1691-92 wrote a little treatise on the differ- 
ential calculus, although this was not published until 1924. 54 The 
first published textbook on the subject appeared in 1696. This was 
by a French disciple of Leibniz, the Marquis de l'Hospital, and the 
title is characteristic of his approach — Analyse des infiniments petits 
pour V intelligence des lignes courbes. This work is based, at least in 
part, on the earlier work of John Bernoulli. 55 Although l'Hospital did 
not in this book discuss the nature of the basic concepts of the cal- 
culus, he played a significant r61e in the popularization of the new 
subject, both through the fact that his text appeared in numerous 
editions 56 and through the influence he exerted in the Journal des 
savants} 7 Through this journal and the Acta eruditorum was created 
an atmosphere of enthusiasm for the differential calculus which led 
to a disregard on the Continent of the method of fluxions. It is inter- 
esting to notice that this parallels the comparative lack of regard for 
Newtonian science, in favor of that of Descartes, until the former 
was popularized in France by Voltaire. In Germany the differential 
calculus was popularized in the philosophical works of Christian 
Wolff, as well as by the mathematical work of the Bernoulli brothers. 
In Italy an enthusiastic interest in the differential calculus was mani- 
fested by Guido Grandi, another of the correspondents of Leibniz and 
author of a number of works on the calculus during the early eight- 
eenth century. 

In spite of the popularity which the calculus of Leibniz enjoyed, 
there was a total lack of clarity and agreement as to the basis of the 
analysis. Voltaire called the calculus "the Art of numbring and meas- 
uring exactly a Thing whose Existence cannot be conceived." 88 The 
indecision which Leibniz had displayed was shared also by his fol- 
lowers. John Bernoulli's little volume on the differential calculus 
begins with the paradoxical postulate that a quantity which is dimin- 
ished or increased by an infinitely small quantity is neither diminished 

64 See Die Differenzialrechnung von Johann Bernoulli. 

M See Enestrom, "Sur la part de Jean Bernoulli dans la publication de 1' Analyse des 
infiniment petits"; cf. Rebel, Der Briefwechsel zwischen Johann (I.) Bernoulli und dent 
Marquis de l'Hospital, pp. 9 ff. 

68 Paris, 1715, 1720, and 1781; Avignon 1768. 

67 Sergescu, "Les Mathematiques dans le 'Journal des Savants,' 1665-1701." 

68 See Letters Concerning the English Nation, p. 152. 



The Period of Indecision 239 

nor increased. 69 In other words, Bernoulli made fundamental the 
omission of differentials of higher orders, 60 rather than the limit con- 
cept. Similarly, in the integral calculus the figure bounded by an 
infinitely small piece of a curve, the ordinates of its end points, and 
the corresponding difference in the abscissas he considered as a par- 
allelogram. 61 Although he thought of a surface as the sum of such 
differentials of area, he did not define the integral as such a sum, as 
had Leibniz, but rather as the inverse of the differential, with the 
addition of a suitably chosen constant 62 — a definition which persisted 
throughout the next century. 

We have seen that Leibniz generally considered his differentials as 
only indefinitely or imcomparably small, but John Bernoulli boldly 
asserted in a letter to Leibniz in 1698 that inasmuch as the number 
of terms in nature is infinite, the infinitesimal exists ipso facto™ 
This assertion he attempted to clarify in a manner recalling the exposi- 
tion of Pascal in which the latter had pointed out that, through the 
reciprocal relationship, the existence of the indefinitely small was 
implied by that of the indefinitely great. Bernoulli sought to apply 
this type of argument to the case of actual infinitesimals: Let the 
infinite series i, i, §, . . . be given. Then if there are ten terms, one- 
tenth exists; if there are a hundred, then a hundredth exists; ... if 
the number of terms is infinite, as is here supposed, the infinitesimal 
exists. 64 Leibniz, in answering, wisely cautioned him that arguments 
concerning the finite need not hold for the infinite, and that further- 
more the infinite and the infinitesimal may be imaginary, even though 
they determine real relationships. 65 John Bernoulli, nevertheless, dis- 
regarded this caveat and persisted in views with respect to the infinitely 
large and the infinitely small which call to mind the earlier work of 
Wallis. 

Of the two famous Bernoulli brothers, John had the greater origi- 
nality and imagination, but James was superior in critical power. 66 

69 John Bernoulli, Die Dijferenlialrechnung, p. 11. 

60 Cf. Leibniz, Mathematische Schriflen, III (Part 1), 366. 

61 John Bernoulli, Die Dijferentialrechnung, p. 11. 

62 John Bernoulli, Die erste Iniegralrechnung, pp. 3, 8, 11-12. 

63 Leibniz, Mathematische Schrijten, III (Part 2), 555. 

64 Ibid., pp. 563 ff.; see also Leibniz and John Bernoulli, Comnterciutn philosophicum et 
malhematicum, I, 400-31. 

68 Commerciiim philosophicum et mathematicum, I, 370. 
M Mach, The Science of Mechanics, pp. 427-28. 



240 The Period of Indecision 

This fact is well illustrated by their respective attitudes toward the 
calculus. Whereas John expressed the positive attitude of Leibniz with 
reference to infinitesimals, his brother James put forth Leibniz's more 
cautious view. He found the use of the infinite not sufficiently con- 
vincing and too far from the opinion of the ancients. 67 He held that the 
infinitely small was not to be thought of as a determined quantity, 
but as a fiction of the spirit — "a perpetual fluxion toward nothing." 

2yy -f- dy 2 
As a consequence, "the ratio is always variable and does 

4yy — dy 2 

not become fixed unless dy is perfectly zero." 68 This view of the 
differential as a variable would associate the calculus with the method 
of limits, but James Bernoulli was unable to express this notion 
clearly because he failed, as had Leibniz, to distinguish between 
independent and dependent variables. In other words, the function 
concept had not yet become primary. 

Although James Bernoulli attempted to avoid the pseudo-infinites- 
imal and maintained that a quantity smaller than any given magnitude 
is zero, 69 he vacillated in his attitude. On occasion he asserted that the 
Euclidean axiom "If equals are taken away from equals, the results 
are equal," need not be absolutely true when incomparably small 
quantities are involved. 70 For this reason he gave warning that in the 
calculus of the infinitely small, one must proceed with caution to 
avoid paralogisms. 71 

Wolff, a follower of Leibniz in mathematics and philosophy whose 
works enjoyed a wide distribution, adopted a modification of the views 
of James Bernoulli which the writings of Leibniz had suggested. He 
thought of the infinitely large and the infinitely small as impossi- 
bilities or as convenient geometrical fictions, useful for discovery, 
which result from a figurative manner of speaking. 72 By calling some- 
thing infinitely large, he said, one simply means that it can exceed 
any number. Similarly, the infinitely small is not really a quantity in 
the strict sense of the word, but rather some sort of imaginary sym- 
bolism as Leibniz had sometimes held. 73 

If some mathematicians followed Wolff in denying the reality of the 

67 Leibniz, Mathematische Schriften, V, 350. 

6? Leibniz, Mathematische Schriften, III (Part 1), 52-56. 

69 Opera, I, 379. 70 Opera, II, 765. n Ibid. 

72 Philosophia prima, sive ontologia, pp. 597-602. n Ibid. 



The Period of Indecision 241 

infinitely small, others took the opposing view. In Italy Guido Grandi 
upheld the existence of absolutely infinite and infinitesimal magni- 
tudes of various orders. Those of first order he defined as quantities 
which bear to any finite magnitude of the same kind a ratio respec- 
tively greater and less than any assignable number. 74 Those of higher 
order were similarly defined in terms of those of lower order. 75 Quan- 
tities differing by less than any assignable magnitude he considered 
equal, because this, he felt, represented only a short way of saying 
what Euclid and Archimedes had meant in their work with in and 
circumscribed figures. 76 As an example of the addition of differentials 
to give a finite magnitude, Grandi referred to the paradoxical result 
l-l + l-l+... = + + ... = i. This, he suggested to 
Leibniz, could be compared with the mysteries of the Christian 
religion and with the creation of the world by which an absolutely 
infinite force created something out of absolutely nothing. 77 

In France, after a period of hesitation, views more bold even than 
those expressed by Grandi were to appear. L'Hospital, the first student 
of John Bernoulli, 78 had presented the views of his teacher in his 
Analyse des infiniments petits; and a little later Pierre Varignon, 
Bernoulli's "best friend in France," 79 worked for the new analysis. 
However, as Nieuwentijdt had opposed the calculus of Leibniz and 
Berkeley that of Newton, so there arose in 1700, in the French 
Academie des Sciences, a lively dispute as to the validity of the 
infinitesimal methods. 80 In this discussion Rolle maintained that the 
new methods led to paralogisms, while John Bernoulli maintained in 
an argumentum ad hominem that Rolle did not understand the cal- 
culus. 81 Varignon attempted to clarify the situation by showing indi- 
rectly that the infinitesimal methods could be reconciled with the 
geometry of Euclid. 82 

In 1727, however, the French author Bernard de Fontenelle, a 
friend of Varignon, could boast that there were no longer two parties 

74 De infiniiis infinitorum et infinite parvorum ordinibus, pp. 22-23. 

75 Ibid., pp. 26 ff. 7 « Ibid., p. 39. 

77 Cf. Leibniz, Mathematisehe Schriften, IV, 215-17; Reiff, Geschichte der unendlichen 
Reihen, p. 66. 

78 Fedel, Der Briefwechsel Johann (i) Bernoulli-Pierre Varignon, p. 3. n Ibid., p. 2. 

80 See a note in Histoire de I' Academie des Sciences, 1701, pp. 87-89. 

81 Leibniz, Mathematisehe Schriften, III (Part 2), 641-^2. 

w Fedel, Der Briefwechsel Johann (i) Bernoulli-Pierre Varignon, p. 25. 



242 The Period of Indecision 

in the Academic He evidenced this fact by publishing his Siemens de 
la geomeirie de I'infini, in which there is no semblance of doubt ex- 
pressed on the subject. The work of Fontenelle displays an absolute 
dogmatism with respect to the infinite. Recognizing that geometry is 
entirely intellectual and independent of the actual description and 
existence of figures, 83 Fontenelle did not discuss the subject from the 
point of view of science or metaphysics, as had Aristotle and Leibniz. 
He objected to regarding the infinite as a mystery and protested 
that Cavalieri was too modest in his treatment. 84 Confidently follow- 
ing Wallis, Fontenelle wrote oo as the last term of the infinite 
sequence 0, 1, 2, 3, . . ., although he realized that the manner in 
which the series goes from the finite to the infinite is inconceivable. 85 
On the basis of this definition of the infinite, Fontenelle went on to 
include in his calculation not only integral powers of °° , but even 
fractional and infinite powers as well, using such symbols as °° * and 
oo °° 3 , and writing such equalities as oo • °° * ~ x = °° °°. 86 As Wallis 

had written the infinitely small as J_, so Fontenelle derived his orders 

oo 

of infinitesimals as the reciprocals of the powers of infinity. The 

differentials dy and dx he held to be magnitudes of the order J_, 

oo 

although he defined these in terms of the characteristic triangle of 
Leibniz. 87 

It is interesting to note that we have found three men — Wallis, 
John Bernoulli, and Fontenelle — who tried in an arithmetic manner 
to derive the infinitely small as the reciprocal of the infinitely large. 
(Pascal had in this manner sought only to relate the indefinitely large 
to the indefinitely small.) Such attempts lacked all semblance of 
mathematical rigor because of the lack at that time of satisfactory 
definitions of cither the infinite or the infinitesimal. These efforts were 
furthermore counter to the general tendency of the time, which found 
the basis of mathematics in geometrical conceptions. Arithmetic had 
not become sufficiently abstract and symbolic to free itself of spatial 
interpretations, for number was still interpreted metrically as a ratio 
of geometrical magnitudes. Descartes had affirmed the identity of 
numerical and geometrical calculations. Newton had said that a num- 

83 Elemms de la geometrie de Vinfini, Preface. M Ibid. 

85 Ibid., pp. 30-31. » 6 Ibid., pp. 40 ff. * Ibid., p. 311. 



The Period of Indecision 243 

ber was a ratio of quantities, and Wolff wrote that a number was 
anything which referred to unity in the same way as one straight line 
to another. 88 The methods of fluxions and differentials were, as a 
result, naturally considered convenient processes for solving geomet- 
rical problems. Although the results were usually expressed in algebraic 
terminology, the bases were sought in the geometry of the ancients, 
rather than in arithmetical conceptions. With the greatest mathe- 
matician of the first half of the eighteenth century, however, a change 
of view — anticipated to some extent by Wallis, Bernoulli, and Fon- 
tenelle — entered the new analysis. 

Leonhard Euler wrote a prodigious number of books and articles 
"promoting the new analysis, organizing it, and putting it on a formal- 
istic basis. Most of his predecessors had considered the differential 
calculus as bound up with geometry, but Euler made the subject a 
formal theory of functions which had no need to revert to diagrams 
or geometrical conceptions. 89 Leibniz had used the word function 
somewhat in our sense, and had boasted that his infinitesimal method 
was not limited to algebraic functions, as was that of Descartes, but 
was applicable to logarithms and exponentials as well. Nevertheless, 
Euler was the first mathematician to give prominence to the function 
concept and to make a systematic study and classification of all the 
elementary functions, together with their differentials and integrals. 

The word function, however, meant for Euler not so much any 
quantity conceived as depending on variables, as an analytic ex- 
pression in constants and variables which could be represented by 
simple symbols. 90 Functionality was a matter of formal representation, 
rather than conceptual recognition of a relationship. The almost 
automatic development of the calculus during the eighteenth century 
was largely the result of this formajistic view, to which the notation 
of Leibniz was so remarkably well adapted. However, the greater the 
success achieved by the differential calculus, the less constrained 
Euler felt to justify his procedures. His views on the bases of the 
subject were elementary in the extreme, resembling somewhat those 

^Elemenla matheseos universalis, I, p. 21; cf. also Pringsheim, "Nombres irrationnels 
et notion de limite," p. 144 n. 

89 See the introduction to his Institutiones calculi differentiate . Opera omnia, Vol. X. 

90 See Brill and Noether, "Die Entwickelung der Theorie der algebraischen Funktionen 
in alterer und neuerer Zeit." 



244 The Period of Indecision 

of Wallis, Taylor, John Bernoulli, and Fontenelle. He felt that the 
notions of the infinitely great and the infinitely small did not hide so 
great a mystery as was commonly thought. An infinitely small or 
evanescent quantity he held to be simply one which will be zero. 91 

This view might well have served as the basis for an interpretation 
in terms of limits, in which the differentials are simply variables ap- 
proaching zero as a limit. Euler, however, did not proceed in this 
manner. Throughout the development of the calculus, the pandemic 
infinitesimal had, at various stages, been taken as a constant quantity 
less than any assignable magnitude. Euler emphatically rejected any 
such notion of mathematical atomism or monadology, inveighing 
against this as a "wretched abuse of the principle of sufficient reason." 92 
He asserted, as had James Bernoulli, that a number less than any 
given quantity must of necessity be zero. 93 The differentials dx and 
dy were therefore simply zero. 94 Although he admitted the existence 
of an infinite number of infinitesimals, as found in the differentials of 
higher orders, these he held were all zero. 95 Leibniz had at one point 
suggested that the differentials could be regarded as qualitative zeros, 
which nevertheless retained by the law of continuity the character 
of the relationships of the finite quantities from which they were 
derived. Euler, in conformity with his formalistic view, held less 
philosophically that the zeros represented by differentials were to be 

distinguished through the recognized fact that the ratio - could, in 

n 
a sense, represent any ratio of finite numbers, ~. 96 Thus for Euler the 

calculus was simply the determination of the ratio of evanescent 
increments — a heuristic procedure for finding the value of the ex- 

pression ~. 97 

91 Opera omnia, X, 69. 

n Letters to a German Princess, II, 61 ; cf . also Opera omnia, X, 67. 

* Opera omnia, X, 69-70. " Ibid., pp. 70-72. 

96 Ibid.; cf . also XI, 5. »« Ibid., X, 70. 

97 Cohen (Das Princip der Infinitesimalmethode und seine Geschichte, p. 96) displays a 
lack of understanding of the limit concept in saying, in connection with Euler's use of 
zeros: "Es ist offenbar, dass er hierein nur der Grenzmethode folgt, dieselbe aber iiber- 
bietet: indem er das Inkrement selbst, nicht den Vorgang, als verschwindend (incrementum 
evanescens) annimmt, als Null." 



The Period of Indecision 245 

In justifying the omission in the differential calculus of infinitesimals 
of higher order, Euler's argument again lacked clarity. He held that, 
in the expression dx =*= dx 2 , the infinitely small quantity dx 2 vanishes 
before dx does, so that for dx = the ratio of dx ± dx 2 to dx will be 
one of equality. 98 At one point he suggested that in omitting terms 
involving differentials, one made allowance for certain errors," as 
Berkeley had maintained; and on occasion he employed difference 
calculus as a practical substitute for the differential calculus. 100 
Euler, however, did not justify the transition from one to the other. 
Leibniz had explained the substitution of inassignables for assignables 
as validated by the law of continuity, but Euler followed Taylor in 
simply substituting zero for the increments. For example, in deter- 
mining the differential of x 2 he allowed x to become x + w . The ratio 
of the increments in x and x 2 was then 1 : 2x + &>. This is always 
different from the ratio 1 : 2x unless w vanishes. Euler therefore sub- 

dx 2 2x 
stituted for w and thus obtained the evanescent ratio — = — in 

dx 1 

much the same manner as Taylor had found the fluxion of a: 2 . 101 

With respect to the infinite, Euler adopted the views of Wallis and 

Fontenelle. Inasmuch as the sum of the series 1 + 2 + 3 + . . . can 

be made greater than any finite quantity, it must be infinite and can 

be represented by the symbol °° , 102 At another point he suggested 

that oo W as a sort of limit between the positive and the negative 

numbers, in this respect resembling the number 0. In a similar manner 

a . . 

he held that the relationship - = °° was to be interpreted as meaning 

that nothing times infinity can result in a finite magnitude. 103 Further- 
more, as — will be infinite because dx = 0, so - — will be infinite of 

dx dx 2 

second order; and, more generally, corresponding to the orders of the 
differentials, there is an infinite number of grades of infinity. 104 If x 
is infinite, he held that between 1 and x 1 ' 000 there are 1,000 grades of 

98 Opera omnia, X, 71. 

99 Cf. Weissenborn, Die Principien der hbhoren Analysis, p. 158. 

100 Cf. Bohlmann, "tjbersicht iiber die wichtigsten Lehrbiicher der infinitesimalrechnung 
von Euler bis auf die heutige Zeit." 

101 Opera omnia, X, 7. 

1W Ibid., p. 69. 1W Ibid., p. 73. m Ibid., p. 75. 



246 The Period of Indecision 

infinity. 105 The lack of care with which Euler handled the infinite is 
evidenced also in his use of divergent series. As Leibniz had suggested 

1 



that 1-1 + 1-1 + ... = |, so Euler held that from 



(1 + 1)' 



= j one could conclude that 1 — 2 + 3— 4 + 5 — ... = £ 



Under a somewhat different point of view Euler added that 1 — 3 + 
5 — 7 + . . . = 0. Numerous similar examples of divergent series are 
to be found in his work. 106 

Inasmuch as Euler restricted himself to well-behaved functions, he 
did not become involved in those subtle difficulties connected with 
the notions of infinity and continuity which were later to make such 
a naive position untenable. Although his views on the fundamental 
principles of the calculus lacked all semblance of the precision and 
rigor which entered mathematics in the following century, the formal- 
ists tendency which his work inaugurated was to free the new analysis 
from all geometrical fetters. 107 It also made more acceptable the arith- 
metic interpretation which was later to clarify the calculus through 
the limit concept which Euler himself neglected. 

While Euler, under the influence of Leibniz and the Bernoullis, was 
working so successfully on the assumption that differentials were 
zeros, his contemporary, Jean le Rond D'Alembert, was promulgating 
the doctrine which was to be elaborated ultimately into that accepted 
at the present time. Although the Newtonian-Leibnizian priority con- 
troversy had estranged British and Continental mathematicians, 
neither group was completely unaware of the views of the other. 
Thus Robins had, in 1 739, criticized Euler for his crude conception of 
the infinitesimal, saying that the error of his ways was due to following 
"that inelegant computist" (John Bernoulli) who instructed him. 108 

In like manner the substance of the Berkeley- Jurin-Robins con- 
troversy was known to mathematicians on the Continent. Buffon, in 
the historical introduction to his translation of Newton's Method of 
Fluxions, criticized Berkeley and Robins for taking exception to some 
of Newton's arguments and warmly espoused Jurin's weak and prolix 
defense as "solid, brilliant, admirable." 109 Although he felt that 

105 Opera omnia, XV, 298. 10S See Opera omnia, XIV, 585 ff. 

m Cf . Merz, A History of European Thought in the Nineteenth Century, I, 103. 

108 Cajori, A History of the Conceptions of Limits and Fluxions, pp. 139-40. 

109 La Methode des fluxions et des suites infinies, pp. xxvii-xxix. 



The Period of Indecision 247 

Robins had criticized Euler and Bernoulli unfairly, 110 Buffon himself 
opposed Euler's views on the infinite and the infinitesimal. He held 
that the sequence 1, 2, 3, . . . had no last term and that the infinitely 
large and the infinitely small were only "privations." 111 Buffon, more 
interested in the natural than the mathematical sciences, did not elab- 
orate this view; but the doctrine that the terms infinitely large and 
infinitely small signified only indefinitely large and indefinitely small 
was expounded more fully by the mathematician D'Alembert, who 
made this the basis of his theory of limits. 

D'Alembert was probably led by two earlier works to regard the 
method of limits as fundamental in the calculus. These books, men- 
tioned in his article on the differentiel in the famous Encyclopedic, 
were Newton's De quadratura curoarum of 1704 and De la Chapelle's 
Institutions de geometric of 1746, a popular text which connected him 
with the adumbrations of the limit idea by Stevin, Gregory of St. 
Vincent, and others. 112 

D'Alembert interpreted Newton's phrase "prime and ultimate 
ratio" not literally, as a first or last ratio of two quantities just spring- 
ing into being, but as a limit. One quantity he called the limit of 
another if the second can approach the first nearer than by any given 
quantity, or so that the difference between them is absolutely inas- 
signable. Properly speaking, however, he felt that the varying quan- 
tity never coincides with, or is equal to, its limit. 113 D'Alembert thus 
agreed essentially with Robins' interpretation of Newton's meaning. 
He applied the same idea to the differential calculus. It will be recalled 
that Leibniz had believed that one could think of differentials as 
inassignable quantities, which by the law of continuity express the 
ultimate relationship between quantities deprived of magnitude. 
However, just as D'Alembert had denied the validity of the concep- 
tion of an ultimate ratio, so also he rejected that of an ultimate 
relationship which can be determined by differentials. He said that 
the basis of the differential calculus, like that of the method of fluxions, 
was to be found in the idea of a limit. "The differentiation of equations 
consists simply in finding the limits of the ratio of finite differences of 

110 Ibid., p. xxix. 

111 Ibid., p. ix. 

m Cajori, "Grafting of the Theory of Limits on the Calculus of Leibniz." 

m Article, "Limite," in Encyclopedic Cf. also Pierpont, "Mathematical Rigor," p. 33. 



248 The Period of Indecision 

two variables included in the equation." 114 This D'Alembert believed 
to be the true metaphysics of the calculus, admitting, incidentally, 
that this was more difficult to develop than were the rules of appli- 
cation. 

Nieuwentijdt had criticized Leibniz's differentials of higher order as 
nonexistant. D'Alembert, however, said that the distinction is irrele- 
vant, for the differential notation is to be considered merely as a 
convenient abridgment or manner of speaking, used to avoid the 
circumlocution necessary in expressing the limit concept. He held that 
no such thing as an infinitesimal existed in its own right. 

A quantity is something or nothing: if it is something, it has not yet van- 
ished; if it is nothing, it has literally vanished. The supposition that there 
is an intermediate state between these two is a chimera. 115 

Just as the calculus of first order differences was explained in terms 
of limits, so D'Alembert defined infinitesimals of second and higher 
orders in a terminology equivalent to that of limiting ratios. His 
first explanation of these quantities lapsed into a phraseology danger- 
ously resembling the naive assertion of Leibniz that dx 2 is defined as 
being to dx as dx is to one, for D'Alembert said: "When one says 
that a quantity is infinitely small with respect to a quantity which is 
itself already infinitely small, this signifies merely that the ratio of 
the first of these quantities to the second is always as much smaller 
than the second quantity as is the latter than a given quantity." 
He added immediately, however, the explanation interpretable in 
terms of limiting ratios, "and that the ratio can be supposed as small 
as we please in imagining the second quantity sufficiently small." 116 
This definition of infinitesimals of higher order indicates the general 
manner in which these quantities are interpreted at the present time, 
but it lacks clarity and resolution. Furthermore, it does not un- 
equivocally state that the limit of the ratio must be zero for infinites- 
imals of higher order. These weaknesses in D'Alembert's explanation 
were corrected early in the following century by Cauchy. Meanwhile 
the interpretation of the infinitely small as a fixed infinitesimal con- 
tinued in spite of D'Alembert's opposition. 

"< Article, "Diflferentiel," p. 977. 

w Melanges de litUrature, d'histoire, et de philosophic, pp. 249-50. m Ibid. 



The Period of Indecision 249 

As D'Alembert had interpreted the word infinitesimal as meaning 
indefinitely small and had defined this in terms of limits, so also did 
he try to clarify the other concept which had bothered mathematicians 
since the Greek period — that of the infinite. He asserted— contrary to 
the view of Fontenelle — that the notion of infinity is really that of the 
indefinitely large and is only a convenient abridgment for the inter- 
pretation in terms of the doctrine of limits. With this understanding, 
he pointed out that one can have orders of infinitely large quantities 
analogous to those of infinitesimals. A line is said to be infinite of the 
second order with respect to another if the ratio of these is greater 
than any given number; it is of the third order if the ratio of the 
product of it by any finite number to the square constructed on the 
other line is greater than any given number. 117 

This interpretation of infinity is, of course, that which Euler had 
in mind when he said that the logarithm of an infinite number is an 
infinite number of lower order than that of any root of an infinite 
number. 118 It corresponds to that which mathematicians use when 
they speak of orders of infinity with respect to functions and is con- 
cerned only with the limits of ratios. It has nothing to do with the 
doctrine of infinite aggregates, which appeared in the late nineteenth 
century and which was necessary rigorously to establish the calculus. 
D'Alembert (like most of his contemporaries) would have been in- 
capable of appreciating the modern notion of the actual infinite. 
In geometry, D'Alembert explained, there was no need to suppose 
the existence of an actual infinite, so that the question of its existence 
did not concern mathematics. Whereas the modern concept of the 
infinite is based upon arithmetic conceptions, D'Alembert 's inter- 
pretations were largely in terms of geometry, even though in the 
chart of the divisions of knowledge, in the preface of the Encyclopedic, 
infinitesimal considerations are entered under, algebra rather than 
geometry. 

Because of his geometrical ideology, D'Alembert's elaboration of the 
limit concept lacked the clear-cut phraseology necessary to make it 
acceptable as a substitute for the infinitesimal interpretation. Thus to 
say with D'Alembert that the secant becomes the tangent when the 
two points are one and that it is therefore the limit of the secant, 119 

UT Ibid., pp. 242 ff. "" Opera omnia, XV, 300. "» Melanges, V, 245-46. 



250 The Period of Indecision 

imposes the necessity of visualizing the process by which two points 
become one, thus leaving the interpretation open to Zeno's criticisms. 

To some mathematicians, at least, D'Alembert's limit concept 
appeared to be enmeshed in as dark a metaphysics as was that of the 
infinitely small. Consequently the majority of textbooks on the cal- 
culus published on the Continent at that time continued to prefer 
the explanations of Leibniz. Of twenty-eight publications appearing 
from 1754 to 1784, fifteen interpreted the calculus in Leibnizian 
terminology, six in terms of limits, four in terms of Euler's zeros, two 
in terms of fluxions, and one (that of Lagrange) in terms of a method 
to be described later. 120 Nevertheless, the limit idea continued from 
time to time to be put forth as the logical manner of interpreting the 
calculus. Hyacinth Sigismund Gerdil, for example, in 1760-61 121 fol- 
lowed D'Alembert in saying that the infinite and the infinitesimal are 
to be excluded from the calculus, their place being taken by limits. 122 

Another supporter was found in A. G. Kastner, the author of a 
popular textbook, Anfangsgrunde der Analysis des Unendlichen, which 
appeared in 1761. In this Kastner said that he made use of Newton's 
method of first and last ratios, although he availed himself also of the 
abridged notation which the differential calculus afforded and which 
he compared, as a sort of ellipsis, with the figures of speech used by 
poets. 123 Following D'Alembert, he denied the existence of both the 
infinitely large and the infinitesimal, although the latter insinuated 
itself into his work nevertheless. In determining differentials, for 
example, Kastner allowed the independent variable z to be given an 
increment e, the function Z of z thereby taking on the increment E. 
"If now e is indefinitely diminished, the limit which the ratio E : e 
approaches indefinitely is called the ratio of the differentials of Z and 
z and the infinitely small quantities e and E are called the differentials 
of z and Z." 124 This lack of that distinction between the increments 
and the differentials which is essential in the interpretation of the 
calculus in terms of the derivative, shows clearly how little even pro- 
fessed proponents of the limit concept could avoid using the infinitely 
small in their explanations. Evidently Kastner did not recognize the 

120 Cajori, "Grafting of the Theory of Limits on the Calculus of Leibniz." 

m Cf. Lagrange, CEuvres, X, 269-70; VII, 598. 

m See Moritz Cantor, Vorlesungen, IV, 643-44. 

123 Anfangsgrunde, Vorrede, pp. xiii-xiv. m Anfangsgrunde, p. 10. 



The Period of Indecision 251 

significance of the admonition of Newton and D'Alembert that the 
ultimate ratio was not a ratio of ultimate quantities. 

That the differentials, rather than the differential quotient, were 
fundamental in his thought is brought out in another connection. In 

determining the differential E of Z = -^r t at the point for which 

a 

z = 0, we should first determine the derivative at this point and 

then multiply this by e, the differential of z. The result is £ = 0. 

Kastner, however, first found the increment E in Z, corresponding 

to the increment e in z, and then substituted 2 = before determining 

E e n 

the limit of the ratio — . His result was E = -^zr v a conclusion which 
e a 

Euler and James Bernoulli had reached by their infinitesimal 

methods. 128 

A number of other mathematicians of the time felt that the infinitely 
small could not furnish a satisfactory basis for the calculus. 126 Most 
of these attempted to substitute some form of the limiting idea, but 
there was one notable exception. Joseph Louis Lagrange displayed a 
skeptical attitude toward the infinitely small, echoing Bishop Berkeley 
in saying that the correctness of the results of the differential calculus 
was to be explained as due to a compensation of errors. 127 However, 
his attitude toward the limit concept was cool also, for he considered 
this involved with metaphysical difficulties. 128 He felt that D'Alem- 
bert's definition of the tangent as the limit of a secant was unsatis- 
factory, inasmuch as after the secant has become the tangent, nothing 
prevents it from continuing as a secant on the other side of the point 
in question. 129 

Furthermore, the method of fluxions did not appeal to Lagrange 
because of the introduction of the irrelevant notion of motion. Euler's 
presentation of dx and dy as failed also to satisfy him because he 
felt that we have no clear and precise idea of the ratio of two terms 
which become zero. 130 Lagrange, as a result, sought a simple algebraic 

m Ibid., p. 13. See also James Bernoulli, Opera omnia, II, 1097; Euler, Opera omnia, X, 
561. 
m See Moritz Cantor, Vorlestmgen, IV, Section XXVI. 

127 Lagrange, (Euvres, VII, 598. See also IX, 17. 

128 (Euvres, VII, 325-26; cf. also III, 443, and IX, 18. 

«• (Euvres, VII, 325. m (Euvres, IX, 17-18. 



252 The Period of Indecision 

method free from the objections found in others. As early as 1759 he 
seems to have been satisfied that he had found this, for he wrote to 
Euler in that year that he believed he had developed the true meta- 
physics of the principles of mechanics and of the differential and 
integral calculus, as far as was possible. 131 Lagrange probably had in 
mind the method 132 which he proposed in a paper of 1772, "Sur une 
nouvelle espece de calcul relatif a la differentiation et a Integration 
des quantites variables." 133 

Here he recalled that Leibniz had pointed out the analogy — now 
called Leibniz's Rule — between differentials of all orders of the prod- 
uct of two variables and the powers of the same order of a binomial 
in these variables, and had also remarked that the same correspond- 
ence subsisted between the negative powers and the integrals. Follow- 
ing this suggestion, Lagrange made use of a similar analogy in con- 
nection with infinite series. The series f(x -f h) = f(x) + f{x)h + 

h 2 
/""(a;)— + . . . had been known at least from the time of Taylor, 

whose name it bears. In this series, the coefficients of the powers of h 
involve the ratios of differentials, or of fluxions. However, the series 
can be derived without reference to these notions. What would be 
more natural, therefore, than to define differentials and fluxions in 
terms of the coefficients of such a series? This procedure would (only 
on the surface, as we know now) obviate the necessity of introducing 
either limits or infinitesimals into the work, and the calculus would 
thus be reduced to simple algebraic operations. This notion of the 
differential and the integral calculus appeared to Lagrange to be 

the clearest and simplest which had yet been given. It is, as one sees, 
independent of all metaphysics and of any theory of infinitely small or 
vanishing quantities. 134 

Lagrange proceeded therefore to make Taylor's series fundamental in 

his work, assuming implicitly that all functions allow of such an 

expansion. The coefficients p, p', . . . g, q' , . . . , of the powers of h, 

k, ... in the Taylor's series for u(x -f h, y -f- k, . . .) he defined as 

the "fonctions derivees" of u. The differential calculus then consisted, 

m (Euvres, XIV, 173; cf. Ill, 443, and VII, 325-28. 

132 See Jourdain, "The Ideas of the 'Fonctions analytiques' in Lagrange's Early Work.'*- 

m See (Euvres, III, 439-76. u« (Euvres, III, 443. 



The Period of Indecision 253 

for Lagrange, in "finding directly and by simple and facile procedures 
the derived functions p,p',..., q, q', - ■ • of the function «"; and the 
integral calculus consisted, inversely, in "determining by means of 
these latter functions the function «." 135 

Lagrange's method is based on the unwarranted supposition that 
every function can be so represented and handled. Moreover, the 
escape from the infinitely large and the infinitely small, as well as 
from the limit concept, is only illusory, inasmuch as these notions 
enter into the critical question of convergence which Lagrange did 
not adequately consider. Furthermore, his method lacks the opera- 
tional suggestiveness and facility which the Leibnizian ideas and 
notation afforded. Logically, however, the Lagrangian definition had 
an advantage in that it sought, as had the work of Euler, to make 
fundamental the formalism of the theory of functions, rather than the 
preconceptions of geometry, mechanics, or philosophy. 

Lagrange has been criticized 136 for giving up, in favor of mathe- 
matical formalism, the "generative" concept which has frequently 
been felt to be the basis of the methods of fluxions and differentials. 
Such criticism is based on a failure to recognize that mathematics is 
most useful when unencumbered by psychological preconceptions. 
Euler had attempted misdirectedly to formalize the Leibnizian con- 
ceptions by making the differentials zeros. D'Alembert had made an 
effort to present a satisfactory idea of the notion of limit, but had 
failed to give the concept a clear and precise formalism which would 
make it logically unequivocal. Lagrange, therefore, sought still 
another mode of presentation, based on the function concept which 
Euler had emphasized and popularized. Incidentally, in so doing he 
focused attention for almost the first time upon the quantity which 
is now the central conception in the calculus — that of the derived 
function, or the derivative, or the differential coefficient (a termin- 
ology reminiscent of Lagrange's method, which was, however, intro- 
duced later by Lacroix). 137 Lagrange, in this connection gave not only 
the name from which the word derivative was adopted, but also the 
notation f'x, modifications of which are still conveniently used. 

138 Ibid. 

m Cohen, Das Princip der Infinilesimalmethode und seine Geschichte, p. 100. 
™ See Bibliotheca Mathematica (3), I (1900), 517 for notes on the origin of these desig- 
nations. 



254 The Period of Indecision 

Newton had emphasized the method of prime and ultimate ratios 
of increments or of fluxions. Although this ratio can be interpreted as 
a single number or quantity, which we now call the derivative, Newton 
seems to have had in mind the idea of this quantity as the ratio of 
increments or of fluxions, or of quantities proportional to these. 
Moreover, although the fluxions themselves can only be defined 
rigorously as derivatives, Newton does not appear to have realized 
this. 

In the differential calculus likewise it is clear that although Leibniz 
realized the significance of the ratio of two infinitesimals, he never 
seems to have thought of this ratio as a single number, but rather as 
a quotient of inassignables, or of assignables proportional to these. 
With D'Alembert's insistence that the differential calculus was to be 
rigorously interpreted only in terms of limits, one comes close to the 
conception of a derivative, but even here there is the lack of the 
notion of a single function, or number, obtained as the limit of a 
single infinite sequence. 

Like Newton and Leibniz, D'Alembert appears to have had in 
mind not a function, but two sides of an equation, the limits of which 
are equal. Kastner likewise displayed the predominance of the idea 
of a ratio. Although he recognized the fundamental significance of the 
limit concept, he nevertheless defined the differential quotient literally 
as a ratio of differentials. 138 In Lagrange's method the term derivative 
can for the first time be applied with strict propriety, for his "fonction 
derivee" is merely a single coefficient of a term in an infinite series 
and is completely divested of any idea of ratio, or limiting equality. 
It is a single quantity or function, and although Lagrange's definition 
was not that which was to be accepted in the end, the notion of a 
derived function may well have aided in making the general accept- 
ance of the current definition possible. 

Although Lagrange's method appeared in the Miscellanea Taurin- 
ensia for 1772, it received little recognition — perhaps as a result of the 
novelty of the ideas and the notation involved — and the search for a 
satisfactory basis for the calculus continued. To encourage efforts 
in this direction the Berlin Academy, of which Lagrange at the time 
was president, in 1784 offered a prize for the best exposition of a clear 
138 Anfangsgriinde, p. 4. 



The Period of Indecision %55 

and precise theory of the mathematical infinity. The prize-winning 
essay was that of Simon L'Huilier: Exposition elementaire des prin- 
cipes des calculs superieurs. This was published in 1787 and appeared 
again (in Latin) in 1795. In this work L'Huilier proposed to show 
that "the method of the ancients, known under the name of Method 
of Exhaustion, conveniently extended, suffices to establish with cer- 
tainty the principles of the new calculus." 139 In conformity with this 
purpose, he modified the method of exhaustion to interpret it in terms 
of limits. Making the limit concept basic in his exposition, L'Huilier 
agreed with D'Alembert that "in the differential calculus it was not 
necessary to pronounce the name differential quantity." 140 

As in modern textbooks, L'Huilier made the differential ratio or 
quotient fundamental, defining this as the limit of the ratio of the 
increment in the function to that in the independent variable. 
L'Huilier regarded this form of presentation of the calculus as a 
development of that of Newton and other English authors, but his 
exposition indicated an advance over their work. He focused atten- 
tion upon a single number (the derivative) as the limit of a single 
variable (the ratio of the increments), rather than upon an ultimate 
ratio of two evanescent quantities, or of two fluxions, or of any two 
quantities which have the same ratio as these. Although he retained 

dy 
the name "differential quotient" and the symbol — to represent this 

ax 

quantity, he insisted that the latter was nothing but a symbol which 

was to be interpreted as a single number. 141 In dealing with differential 

d 2 y 
quotients of higher orders he again warned that the symbol — was 

not to be broken up as a quotient. This is in striking contrast with the 

work of Newton and Leibniz, in which fluxions and differentials of 

any order were regarded as having a significance independent of the 

ratio or equation in which they entered. The differential quotient of 

L'Huilier was a single number or function, equivalent to Lagrange's 

derived function, and represented essentially the present conception 

of the derivative. 

Although L'Huilier 's definition of the differential quotient is in 

139 Exposition iUmentaire, p. 6. 

"° Ibid., p. 141. m IMd., p. 32. 



%56 The Period of Indecision 

most respects that to be found in present day elementary textbooks 
on the calculus, he does not seem to have been aware of the possible 
difficulties involved in the limit concept. He had avoided the mysti- 
cism of the infinitesimal, the vagueness of the ultimate ratio, and the 

inanity of the symbol -; but he failed to appreciate that the subtlety 

of the limit concept was to make an extremely careful definition 
essential. L'Huilier was dealing only with very simple functions, so 
that he was unaware of the inadequacies of his presentation. His 
variable was always less than or greater than his limit. "Given a 
variable quantity always smaller or greater than a proposed constant 
quantity; but which can differ from the latter by less than any pro- 
posed quantity however small; this constant quantity is called the 
limit in greatness or in smallness of the variable quantity." 142 The 
variable could not oscillate, as it may under our more general view. 

More seriously, L'Huilier fell into an error which is suggested by 
the vague idea of uniformity expressed in the law of continuity of 
Leibniz. He said that "if a variable quantity at all stages enjoys a 
certain property, its limit will enjoy this same property." 143 That 
this notion persisted also in the nineteenth century is apparent from 
the statement of William Whewell: "The axiom . . . that what is 
true up to the limit is true at the limit, is involved in the very con- 
ception of a limit." 143a The falsity of this doctrine is immediately 
apparent from the fact that irrational numbers may easily be 
defined as the limits of sequences of rational numbers, or from 
the observation that the properties of a polygon inscribed in a 
circle are not those of the limiting figure — the circle. The mistake 
of L'Huilier in this connection was probably the result of his 
failure to see that the limit concept was to be identified with 
the nature of infinite converging sequences and with the question 
of the nature of real numbers and the continuum. The inadequacy of his 
conception of number may be further inferred from the fact that he felt 
it necessary to distinguish between limiting values and limiting ratios. 

10 Ibid., p. 7. 

143 Exposition ilimentaire, p. 167. Cajori ("Grafting of the Theory of Limits on the Calcu- 
lus of Leibniz"), apparently unaware of this statement, has said that this principle was used 
but not stated in the eighteenth century. 

143 * History of Scientific Ideas, I., 152. 



The Period of Indecision 257 

Although L'Huilier correctly sought the basis of the calculus in 
the limit concept, his exposition of this was an oversimplification of 
what was later realized to be a very difficult question. Following 
D'Alembert in regarding the infinite from the point of view of mag- 
nitude rather than of aggregation, he denied the existence of an 
actually infinite quantity because he felt that its acceptance would 
lead to such "contradictions" as t» + n = °° - n. He consequently 
maintained that he had shown the calculus to be independent of all 
idea of the infinite, whether large or small. In this he failed to realize 
that the whole theory of limits is based, in the last analysis, upon 
that of infinite aggregates. This fact was not clearly recognized until 
the following century. 

L'Huilier's monograph was not widely read, nor were his views 
generally accepted at the time. The indecision as to the true basis of 
the calculus remained as acute as before. As a result, there appeared 
in 1797 what was perhaps the most famous attempt to clear up the 
difficulties in the situation: the Reflexions sur la metaphysique du 
calcul infinitesimal, by L. N. M. Carnot, the remarkable soldier, 
administrator, and mathematician to whom the French Assembly 
gave the title "L'Organizateur de la Victoire." Carnot's work enjoyed 
a truly remarkable popularity, appearing in numerous editions and 
several languages from that time until quite recently. 144 

In view of the lack of clarity and uniformity in the then-current 
expositions of the calculus, Carnot wished to make the theory rigidly 
precise. Considering the many conflicting interpretations of the sub- 
ject, he sought to know "in what the veritable spirit of the infinitesimal 
analysis consisted." 145 In his selection of the unifying principle, how- 
ever, he made a most deplorable choice. He concluded that "the true 
metaphysical principles of the Infinitesimal Analysis . . . are never- 
theless . . . the principles of the compensation of errors," 146 as Berk- 

144 The first edition of Paris, 1797, was followed by an enlarged second edition at the 
same place in 1813. Unless otherwise stated, references in this work are to this second 
edition. The first edition was translated into English by W. Dickson in the Philosophical 
Magazine, VIII (1800), 222-40; 335-52; IX (1801), 39-56. The second edition appeared 
in an English translation by W. R. Browell at Oxford, in 1832, as Reflexions on the Meta- 
physical Principles of the Infinitesimal Analysis. Other French editions appeared in Paris 
in 1839, 1860, 1881, and 1921 (2 vols.) A Portuguese translation appeared at Lisbon in 
1798, a German one at Frankfurt a. M. in 1800, and an Italian one at Pavia in 1803. 

146 Reflexions sur la mitaphysique dn calcul infinitesimal, p. 1. 146 Browell trans., p. 44. 



258 The Period of Indecision 

eley and Lagrange had suggested. In his expansion of this view, he 
reverted substantially to ideas which Leibniz had expressed. He held 
that to be certain that two designated quantities are rigorously equal, 
it is sufficient to prove that their difference cannot be a "quantite 
designee." 147 Paraphrasing Leibniz, Carnot said further: that for any 
quantity one may substitute another which differs from it by an 
infinitesimal; 148 that the method of infinitesimals is nothing more than 
that of exhaustion reduced to an algorithm; 149 that "quantites inap- 
preciables" are merely auxiliaries which are introduced, like imaginary 
numbers, only to facilitate the computation, and which are eliminated 
in reaching the final result. 150 

Carnot even echoed the favorite explanation of Leibniz in terms of 
the law of continuity. He held that one can envisage the infinitesimal 
analysis under two points of view, according as the infinitesimals are 
taken to be "quantites effectives" or are regarded as "quantites 
absolument nulles." (Leibniz, however, had not admitted them as 
absolutely, but only relatively, zero.) In the first case, he felt that the 
calculus was to be explained upon the basis of a compensation of 
errors: "imperfect equations" were to be made "perfectly exact" by 
the simple expedient of eliminating the quantities whose presence 
occasioned the errors; 151 in the latter case he considered the calculus 
an "art" of comparing vanishing quantities with each other in order 
to discover from these comparisons the relationships between the pro- 
posed quantities. 152 To the objection that these vanishing quantities 
either are or are not zero, Carnot responded that "what are called 
infinitely small quantities are not simply any null quantities at all, 
but rather null quantities assigned by a law of continuity which 
determines the relationship." 153 This explanation is strikingly like that 
given by Leibniz about a century earlier. 

Along with his establishment of the "true metaphysics" of the 
infinitesimal calculus, Carnot proceeded to show that the diverse 
views of the subject were essentially reducible to this same basis. 

147 Riflexions sur la mitapkysique du calcul infinitesimal, p. 31. 

148 /Wrf., p. 35. i« Ioid} p> 39 
160 Ibid., pp. 38-39. "Les mathematiques ne sont-elles pas remplies de pareilles enigmes?" 

said Carnot, in this connection. 
151 Cf. Dickson's translation, p. 336. 
162 Riflexions sur la mitapkysique du calcul infinitesimal, p. 185. 1M Ibid., p. 190. 



The Period of Indecision 259 

In demonstrating this, he pointed out that the method of exhaustion 
made use of analogous systems of known auxiliary quantities. Newton's 
method of prime and ultimate ratios was similar, except that in it the 
lemmas freed the work from the need for the argument by a double 
reductio ad absurdum. Carnot felt that the methods of Cavalieri and 
Roberval also were admittedly corollaries of the method of exhaus- 
tion. 154 Descartes' method of undetermined coefficients, he held, 
touched the infinitesimal analysis closely, the latter being but "a 
felicitous application" of the former! 155 The method of limits he recog- 
nized to be not different from that of first and last ratios, so that it 
was likewise a simplification of the method of exhaustion. 156 Further- 
more, these methods all lead to the results of infinitesimal analysis, 
but by a difficult and circuitous route. 157 Lagrange's method he like- 
wise saw linked to the infinitesimal calculus, in that it neglects the 
other terms of the infinite series. 158 The divers points of view he 
therefore felt, somewhat as had L'Hurlier, to be merely simplifications 
of the method of exhaustion which were effected by reducing this to a 
convenient algorithm. Inasmuch as the infinitesimal method combined 
the facility of the procedures of approximate calculation with the 
exactitude of the results of ordinary analysis, he saw no point in 
attempting, under the guise of greater rigor, to substitute for it any 
less natural method. 159 

Although Carnot's work was widely read, it can hardly be said to 
have led to a clearer understanding of the difficulties inherent in the 
new analysis. Although he realized that differentials were to be denned 
somewhat as variables, anticipating, to a certain extent, the view of 
Cauchy, he was unable to give suitable definitions to these because, 
like Leibniz, he thought in terms of equations rather than of the 
function concept which led Cauchy to make the derivative funda- 
mental. Furthermore, Carnot, one of a school of mathematicians who 
emphasized the relationship of mathematics to scientific practice, 160 
appears, in spite of the title of his work, to have been more concerned 
about the facility of application of the rules of procedure than about 
the logical reasoning involved. In this respect his work resembles 

154 Ibid., pp. 139-40. I56 Ibid., pp. 150-51. 1M Ibid., p. 171. 

157 Ibid., p. 192. i» Ibid., pp. 194-97. » Ibid., pp. 215-16. 

160 Merz, A History of European Thought in the Nineteenth Century, II, 100-1. 



260 The Period of Indecision 

that of Leibniz, whose explanations of differentials he so largely 
paraphrased, and whose method he defended with almost polemic 
warmth. 

Mathematicians on the Continent had been in essential agreement 
that pragmatically the differential calculus offered the best method 
of procedure; but it was precisely on the logic of the matter that 
they were at variance. On the latter point Carnot was of little assist- 
ance, for after pointing out what most men had long realized — that 
all of the methods of the new analysis were essentially related — he 
proceeded to make basic the infinitesimal system, the one which was 
logically, perhaps, the weakest of all. In so doing he pointed toward 
a view diametrically opposed to that which D'Alembert 161 and 
L'Huilier had indicated and along which the rigorous development 
was ultimately to proceed, following the work of Cauchy. 162 

The year 1797, in which the first edition of Carnot's work was 
published, saw the appearance also of the famous work of Lagrange, 
Theorie des fonctions analytiques, contenant les principes du calcul 
differentiel, degages de toute consideration d'infiniment petits ou d'evan- 
ouissans, de limites ou de fluxions, et redmts a V analyse algebrique des 
quantites finies. This book developed with care and completeness the 
characteristic definition and method in terms of "fonctions derivees," 
based upon Taylor's series, which Lagrange had proposed in 1772. 
In it the author gave not only an attempted proof of the incorrect 
theorem that every continuous function may be so expanded, but 
also the determination of the "fonctions derivees" (or derivatives) of 
the elementary functions, and numerous applications to geometry and 
mechanics. Carried along by the authority of Lagrange's great repu- 
tation, the method now enjoyed a short-lived period of comparative 
success. As had been the case with Maclaurin's treatise on fluxions, 
mathematicians praised the new method highly, 163 although they 
seldom used it. They explained that inasmuch as the notations were 

181 Mansion (Esquisse de I'kistoire du calcul infinitesimal, p. 290, n.) has misdirectedly 
characterized Carnot's ideas as a development of those of D'Alembert. 

1,2 The assertion (See Smith, "Lazare Nicholas Marguerite Carnot," p. 189) that Carnot 
"paved the way for Cauchy's notable memoir" is justified only in the most general sense. 

143 Cf. the review in Monthly Review, London, N. S., XXVIII (1799), appendix, pp. 481- 
99; also Valperga-Caluso, "Sul paragone del calcolo delle funzioni derivate coi metodi 
anteriori"; also the set of monographs by Froberg, and others, called De analytica calculi 
differentialis et integraiis theoria, inventa a eel. La Grange. 



The Period of Indecision 261 

less convenient and the calculations involved more embarrassing than 
those found in the methods of the ordinary differential and integral 
calculus, it was sufficient that one were assured through Lagrange's 
method of the legitimacy of other more expeditious methods. This 
seems to have been the view of Lagrange himself, for side by side 
with his method of derived functions he continued throughout his life 
to employ the notation of differentials. 

Most of the objections to the method of Lagrange were based upon 
the inconveniences of the notation and the operations, but before long 
doubts began to arise as to the correctness of the principle that all 
continuous functions could be expanded in Taylor's series. It was 
pointed out 164 that such an expansion was possible only for the more 
simple functions, and that consequently the method was of limited 
applicability. Furthermore, Sniadecki, a Pole, correctly explained that 
the method was fundamentally identical with the limit method. 165 
However, the views of convergence, continuity, and function held at 
the time were not sufficiently definite to permit a deeper clarification 
of these ideas. 

An interesting but somewhat misdirected tirade against Lagrange's 
point of view was delivered by another Polish mathematician, Hoene 
Wronski. An eager devotee of the differential method of Leibniz and 
of the transcendental philosophy of Kant, he protested with some 
asperity against the ban on the infinite in analysis which Lagrange 
had wished to impose. He criticized Lagrange not so much for the 
absence of logical rigor in his free manipulation of infinite series — 
although he did pertinently ask where Lagrange got the series f(x + i) 
= A + Bi + Ci 2 + Di 3 -f . . . with which he opened his proof of 
Taylor's series — as for his lack of a sufficiently broad view. Wronski 
believed that modern mathematics is to be based on the "supreme 
algorithmic law" Fx ~ A Qo + ^4iS2i + A& 2 + A 3 Q, 3 -f . . . , where 
the quantities fio, Qi, fl2, 83, • • . are any functions of the variable x. 
Being the supreme law of mathematics, the irrecusable truth of this 
law he held to be not mathematically derived, but given by trans- 
cendental philosophy. 166 

164 See Dickstein, "Zur Geschichte der Prinzipien der Infinitesimalrechnung"; cf. also 
Lacroix, Traiti du calcid, 2d ed., Ill, 629-30. 
166 Dickstein, "Zur Geschichte der Prinzipien der Infinitesimalrechnung." 
1,8 Refutation de la theorie des fonctions analytiques, Vol. IV of Wronski's (Euvres. 



262 The Period of Indecision 

Wronski was, of course, correct in saying that the method of 
derived functions was too restricted, in that it was limited only to 
certain functions which could be so expanded. However, his general 
views on the calculus were far from those accepted at the present 
time. Whereas Lagrange had attempted to give a formal logical 
justification of the subject, Wronski asserted that the differential 
calculus constituted a primitive algorithm governing the generation of 
quantities, rather than the laws of quantities already formed. Its 
propositions he held to be expressions of an absolute truth, and the 
deduction of its principles he consequently regarded as beyond the 
sphere of mathematics. The explication of the calculus by the methods 
of limits, of ultimate ratios, of vanishing quantities, of the theory of 
functions, he felt constituted but an indirect approach which pro- 
ceeded from a false view of the new analysis. In seeking to examine 
the principles of the subject by purely mathematical means, Wronski 
believed geometers were simply wasting time and effort. He called 
upon them to give up "that servile imitation of ancient geometers in 
avoiding the infinite and that which depends upon it." 167 

In his own work Wronski made a more uncritical use of the infinite 
than had even Wallis and Fontenelle. He said, for example, that "the 
absolute meaning of the number t was given by the expression . . . 

4 oo f ] 

^r== J (1 + V - i)i - (l _ V - i)i I ."i68 Perhaps because of the 

novelty of his notation, as well as of this bizarre use of the symbol <*> , 
Wronski did not exert a strong influence upon the development of 
the calculus. The mathematics of the time was about to accept what 
he opposed: the rigorous logical establishment of the calculus upon the 
limit concept. Nevertheless, the work of Wronski represents an ex- 
treme example of a view which we shall find recurring throughout the 
nineteenth century. In regarding the calculus as a means of explaining 
the growth of magnitudes, followers of this school of thought were 
to attempt to retain the concept of the infinitely small, not as an 
extensive quantity but as an intensive magnitude. Mathematics has 
excluded the fixed infinitely small because it has failed to establish 
the notion logically; but transcendental philosophy has sought to 
preserve primitive intuition in this respect by interpreting it as 
w Ibid., pp. 39-40. "• Ibid., p. 69. 



The Period of Indecision 263 

having an a priori metaphysical reality associated with the generation 
of magnitude. 

Lagrange's Theorie des fonctions was only one, but by far the most 
important, of many attempts made about this time to furnish the cal- 
culus with a basis which would logically modify or supplant those 
given in terms of limits and infinitesimals. Condorcet, Arbogast, 
Servois, and others put forth methods similar to that of Lagrange. 169 
Condorcet, as early as 1786, had begun a work on the calculus, based 
on series and finite differences, but this was interrupted by the Revolu- 
tion and did not appear. 170 

In 1789 L. F. A. Arbogast presented to the Academie des Sciences a 
"true theory of the differential calculus" along the lines of Lagrange 
and Condorcet. This work, published in 1800 under the title Du 
calcul des derivations, sought to establish the principles of the calculus 
independently of limits and the infinitely small, and with the sim- 
plicity and certitude found in ordinary algebra. 171 Arbogast assumed, 
as had Lagrange, that the function F( a + x) could be expanded in a 
series of powers of x, and then showed that the coefficients in this 
could be identified with the more familiar differential quotient. 172 

These new methods resemble the attempts which had been made 
in England, following the Berkeley- Jurin-Robins dispute, to give 
arithmetical procedures for the calculus. They serve to indicate, as 
did the British controversy, a lack of satisfaction, at the time, with 
the methods of limits, fluxions, and infinitesimals. The full title of 
Lagrange's Theorie des fonctions indicates his discontent with these 
methods. F. J. Servois, in his Essai sur un nouveau mode d 'exposition 
des principes du calcul differentiel, which appeared in 1814, 173 expressed 
himself more strongly. He called the method of limits a "gothique 
hypothese" 174 and that of differentials a "strabisme infinitesimal." 176 

IM For references to this work, see Dickstein, op. cit.; Moritz Cantor, Vor lesungen, IV, 
Section XXVI; and Lacroix, Traitt du calcul differential et du calcul integral, I, 237-48, 
and preface. 

170 See Lacroix, Traits du calcul, I, xxii-xxvi. 

171 See Zimmermann, A rbogast als Mathematiker und Historiker der Mathematik, pp. 44-45 ; 
cf. also Arbogast, Du calcul des derivations, Preface; and Lacroix, TraiU du calcul, Preface. 

172 Du calcul des derivations, pp. xii-xiv; cf. also p. 2. 

173 This appeared in the Annates de Mathematiques, V (1814-15), pp. 93-141, and was 
also published separately in Nimes, 1814. References given in this work are to the Nlmes 
publication. 

»< Essai, p. 56. ire /&#., p , 65. 



264 The Period of Indecision 

The introduction of the idea of infinity he objected to as useless; and, 
with reference to the views which Wronski had expressed on the cal- 
culus in his Refutation of Lagrange, he asserted that he "had well 
foreseen, on reading Kant, that geometers would sooner or later be 
the object of the cavils of his sect." 176 Servois tried, in his turn, to 
establish the calculus on a combination of finite differences and 
infinite series, in which the idea of limits was, as in the case of the 
method of Lagrange, ostensibly eliminated by a disregard of the 
essential question of convergence. 

Further attempts along this line continued for sometime: in 1821 
in Eine neue Method efiir den Infinitesimalkalkul of Georg von Buquoy ; 
in 1849 in the Essai sur la metaphysique du calcul integral of C. A. 
Agardh; and as late as 1873 in an article, "Exposition nouvelle des 
principes de calcul differentiel," by J. B. Brasseur. 177 These invariably 
depended upon expansions in series to avoid "any metaphysical no- 
tions," and in this respect they resembled closely the earlier efforts of 
Lagrange, Condorcet, Arbogast, and Servois. However, at almost pre- 
cisely the period of the work of Servois a tendency toward rigor in 
series was setting in. This made clear the basic weakness of all such 
attempts to avoid the method of limits through an uncritical formal 
manipulation of infinite series. The final formulation of the calculus 
was not destined to be in terms of one of these new methods which 
were developed, but was to be based on one of the very notions they 
sought to avoid — that of a limit. 

In the year 1797, in which Carnot attempted a concordance of all 
the systems of the calculus and in which Lagrange tried seriously to 
establish his method, there was published the first volume of perhaps 
the most famous and ambitious textbook on the subject which had 
appeared up to that time — Traite du calcul differentiel et du calcul 
integral of Lacroix. Lacroix declared in the preface of this work that 
it had been the new method of Lagrange which had inspired him to 
compose a treatise on the calculus which should have as a basis the 
luminous ideas which this method had substituted for the infinitely 
small. His text, however, indicates well the indecision of the period, 
for in spite of the professed aim of the work, the foundation of the 
subject remained somewhat in doubt. Lacroix interpreted the method 
174 Ibid., p. 68. m See Bibliography below, for full citations of these works. 



The Period of Indecision 265 

of series of Lagrange in terms of the limits of D'Alembert and L'Huilier. 
In this respect, however, he obscured the significance of this relation- 
ship by speaking of the limit of divergent series, and by following 
Euler in the study of such infinite series asl — 1+2 — 6-f-24 — 
120 + . . . m Furthermore, he did not share with Lagrange the sus- 
picion that the method of Leibniz was based on a false idea of the 
infinitely small. Lacroix admittedly made use of infinitesimals. 179 
Although he accepted the metaphysics of Lagrange, he adopted the 
differential notation of Leibniz. 180 This fact at times confused his 
thought and led him, as it had Euler, to regard the differential coeffi- 
cient (as Lacroix called it) as a quotient of zeros. 181 

The lack on the part of Lacroix of critical distinction in dealing 
with the methods of Newton, Leibniz, and Lagrange gave to his work 
an appearance somewhat resembling the attempt of Carnot to demon- 
strate the congruity of the numerous representations of the calculus. 
The mathematician and astronomer Laplace praised this attitude on 
the part of Lacroix, saying that such a rapprochement of methods 
served as a mutual clarification, the true metaphysics probably being 
found in what they had in common. 182 It appears, however, that at 
that time at least it was deplorable, for it led to a confusion of thought 
on the subject just when logical precision was most needed. However, 
in his more popular work of 1802 — the Traite elementaire, an abridg- 
ment of his larger treatise — Lacroix omitted the method of Lagrange 
and made the explanation in terms of limits basic, although again 
with a lack of rigor which made interpretations in terms of infinites- 
imals possible. The success of this work, which went through many 
editions (the ninth French edition appearing in 1881) and was trans- 
lated into several languages, led to other texts of the same type; and 
it was largely through these that the method of limits became familiar, 
if not rigorous. It was through such texts that the Leibnizian notation 
and the doctrine of limits supplanted in England the method of 
fluxions and interpretations which had already become hopelessly 
confused with the infinitely small. 

The year 1816, in which Lacroix's shorter work was translated into 

m Traite du calcul, HI, 389. 

m Ibid., I, 242. "> Ibid., p. 243. ™ Ibid., I, 344. 

m Letter to Lacroix, published in Le Nouveau Traits du calcul difftrentid, 2d ed., preface, 
p.XLX. 



266 The Period of Indecision 

English, "marks an important period of transition," 183 because it 
witnessed the triumph in England of the methods used on the Conti- 
nent. This particular point in the history of mathematics marks a 
new epoch for a far more significant reason, for in the very next year 
there appeared a work by Bolzano — Rein analytischer Beweis des 
Lehrsatzes dass zwischen je zwei . Wertlien, die ein entgegengesetztes 
Resultat gewahren, wenigstens eine reelle Wurzel der Gleichung liege — 
which indicated the rise of the period of mathematical rigor in all 
branches of the subject. 184 In the calculus the new attitude resulted 
in the logical establishment of the higher analysis upon the limit 
concept, and thus brought to an end the period of indecision which 
had begun with the inventions of the method of fluxions and of the 
differential calculus. 

m Cajori, A History of the Conceptions of Limits and Fluxions, pp. 270-71. 
m See Pierpont, "Mathematical Rigor," pp. 32-34. 



VII. The Rigorous Formulation 

THE OBJECTIONS raised in the eighteenth century to the 
methods of fluxions, of prime and ultimate ratios, of limits, of 
differentials, and of derived functions were in large measure un- 
answered in terms of the conceptions of the time. The arguments were 
in the last analysis equivalent to those which Zeno had raised well 
over two thousand years previously and were based on questions of 
infinity and continuity. The proponents of all methods except that of 
differentials, however, protested that they had no need to invoke the 
notion of the infinite and disregarded entirely that of continuity. The 
advocates of the differential calculus, in turn, although attempting to 
justify its procedures in terms of these concepts, were quite unable 
to furnish logically consistent explanations of them. By most mathe- 
maticians these were considered to be metaphysical ideas and so to lie 
beyond the realm of mathematical definition. 

It is interesting, in looking back, to see that the methods most 
adverse to the introduction into mathematics of the notions of infinity 
and of continuity were precisely those which made this introduction 
possible. The method of limits, which at that time appeared to lead 
to neither infinity nor the law of continuity, was to furnish the logical 
basis for these; and the method of Lagrange, which was developed in 
order to avoid these difficulties, was to bring up questions which 
pointed the way toward their solution. We have seen that in the early 
nineteenth century critics of Lagrange began to question the validity 
of his principle that a continuous function can always be expressed, 
by means of Taylor's theorem, as an infinite series. They began to ask 
what was meant by a function in general and by a continuous function 
in particular, and to criticize the almost indiscriminate use of infinite 
series. Lagrange had made a start in the direction of greater care in 
the use of series when he pointed out that one must consider the 
remainder in every case. This warning serves to indicate, perhaps, 
why he thought he had avoided the use of the infinite and the infi- 
nitely small. Like Archimedes, he evidently did not consider the series 
as continued to infinity, but only to a point at which the remainder 



268 The Rigorous Formulation 

was sufficiently small. In the nineteenth century, however, the con- 
cept of infinity was to become basic in the calculus through the use 
of infinite series and infinite aggregates. 

One of the pioneers in the matter of greater rigor in the funda- 
mental conceptions of the calculus — in its arithmetization and in the 
careful study of the infinite — was the Bohemian priest, philosopher, 
and mathematician, Bernhard Bolzano. 1 In 1799 Gauss had given a 
proof of the fundamental theorem of algebra — that every rational, 
integral algebraic equation has a root — using considerations from 
geometry. Bolzano, however, wished a proof involving only con- 
siderations derived from arithmetic, algebra, and analysis. As Lagrange 
had felt that the introduction of time and motion into mathematics 
was unnecessary, so Bolzano sought to avoid in his proofs any con- 
siderations derived from spatial intuition. 2 

This attitude made necessary, in the first place, a satisfactory 
definition of continuity. The calculus may, indeed, be thought of as 
having arisen from the Pythagorean recognition of the difficulty 
involved in attempting to substitute numerical considerations for 
supposedly continuous geometrical magnitudes. Newton had avoided 
such embarrassment by appealing to the intuition of continuous 
motion, and Leibniz had evaded the question by his appeal to the 
postulate of continuity. Bolzano, however, gave a definition of con- 
tinuous function which, for the first time, indicated clearly that the 
basis of the idea of continuity was to be found in the limit concept. 
He defined a function f(x) as continuous in an interval if for any 
value of x in this interval the difference f(x + Ax) — f(x) becomes 
and remains less than any given quantity for Ax sufficiently small, 
whether positive or negative. 3 This definition is not essentially differ- 
ent from that given a little later by Cauchy and is fundamental in the 
calculus at the present time. 

In presenting the elements of the calculus, Bolzano realized clearly 
that the subject was to be explained in terms of limits of ratios of 

1 See Stolz, "B. Bolzano's Bedeutung in der Geschichte der Infinitesimalrechnung." 
Although Bolzano was born and died at Prague, Quido Vetter ("The Development of 
Mathematics in Bohemia," p. 54) has called him a Milanese, inasmuch as his father was 
a native of northern Italy. 

2 Rein analytiscker Beweis, pp. 9-10. 

3 See Bolzano's Schriften, I, 14; cf. also Rein analytiscker Beweis, pp. 11-12. 



The Rigorous Formulation 269 

finite differences. He defined the derivative of F(x) for any value of 

Fix -f- Ax) — F(x) 

x as the quantity F'(x) which the ratio approaches 

Ax 

indefinitely closely, or as closely as we please, as Ax approaches 

zero, whether Ax is positive or negative. 4 This definition is in essence 

the same as that of L'Huilier, but Bolzano went further in explaining 

the nature of the limit concept. Lagrange and other mathematicians 

had felt that the limit notion was bound up with a quotient of evan- 

dy 
escent quantities or of zeros. Euler had explained — as a quotient of 

ax 

zeros, and in this respect Lacroix tended to follow him. Bolzano, how- 
ever, emphasized the fact that this was not to be interpreted as a 
ratio of dy to dx or as the quotient of zero divided by zero, but was 
rather one symbol for a single function. 6 He held that a function has 

no determined value at a point if it reduces to -. However, it may 

have a limiting value as this point is approached, and he correctly 

indicated that, by adopting the limiting value as the meaning of -, 

the function may be made continuous at this point. 6 

Ever since the invention of the calculus it had been felt that, 
inasmuch as the subject was bound up with motion and the growth 
of magnitudes, the continuity of a function was sufficient to assure 
the existence of a derivative. In 1834, however, Bolzano gave an 
example of a nondifferentiable continuous function. This is based upon 
a fundamental operation which may be described as follows: Let 
PQ be a line segment inclined to the horizontal. Let it be halved by 
the point M, and subdivide the segments PM and M Q into four 
equal parts, the points of division being Pi, Pi, ^3, and Q 1} Q 2 , Q 3 . 
Let Pz be the reflection of P% in the horizontal through M, and let 
Qz be the reflection of Q 3 in the horizontal through Q. Form the 
broken line PP 3 'MQ 3 'Q. Now apply to each of the four segments of 
this broken line the fundamental operation as described above, ob- 
taining in this manner a broken line of 4 2 segments. Continuing 

* Schriften, I, 80-81 ; cf . Paradoxien des Unendlichen, p. 66. 

* Paradoxien des Unendlichen, p. 68. • Schriften, pp. 25-26. 



270 The Rigorous Formulation 

this process indefinitely, the broken-lined figure will converge toward 
a curve representing a continuous function nowhere differentiable. 7 

This illustration given by Bolzano might have served in mathe- 
matics as the analogue of the experimentum cruris in science, showing 
that continuous functions need not, in spite of the suggestions of 
geometrical and physical intuition, possess derivatives. However, 
because the work of Bolzano did not become known at the time, such 
a role was reserved for the famous example of such a function given by 
Weierstrass about a third of a century later. 8 

Lagrange had held that his method of series avoided the necessity 
of considering the infinitesimals or limits, but Bolzano pointed out 
that in the case of infinite series it is necessary to consider questions 
of convergence. These are analogous to limit considerations, as is 
obvious from the statement of Bolzano that if the sequence Fi(x), 
F 2 (x), F 3 (x), . . . , F n (x), . . . , F n + r (x), ... is such that the differ- 
ence between F n (x) and F n + r (x) becomes and remains less than any 
given quantity as n increases indefinitely, then there is one and only 
one value to which the sequence approaches as closely as one pleases. 9 
This fundamental proposition of Bolzano will be seen later to have 
also a significance with respect to the general definition of real number 
and the arithmetical continuum. 

Bolzano felt, in spite of the paradoxes presented by the notions of 
space and time, that any continuum was to be thought of as ulti- 
mately composed of points. 10 His view in this respect resembles that 
of Galileo, to whom he referred in this connection. 11 Although he 
denied the existence of infinitely large and infinitely small magni- 
tudes, he maintained, with Galileo, the possibility of an actual infinity 
with respect to aggregation. He remarked, with respect to such 
assemblages, the paradox which Galileo had pointed out: that the 
part could in this case be put into one-to-one correspondence with the 
whole. The numbers between and 5, for example, could be paired 
with those between and 12. 12 Bolzano's views on the infinite are 
substantially those which mathematicians have adopted since the 

7 See Kowalewski, "Uber Bolzanos nichtdifferenzierbare stetige Function." 
8 Waismann (Einfuhrung in das mathematische Denken, p. 122) mistakenly represents 
Weierstrass as the first to give such an example. 

9 Rein anaiytischer Beweis, p. 35. 10 Paradoxien des Unendlichen, pp. 75-76. 

11 Ibid., pp. 89-92. u Ibid., pp. 28-29. 



The Rigorous Formulation 271 

time of Cantor, except that what Bolzano had considered as different 
powers of infinity have been discovered to be of the same power. 
Bolzano, however, sought to prove the existence of the infinite upon 
theological grounds, whereas later in the century the property which 
he and Galileo had regarded simply as a paradox was made by 
Dedekind, in his clarification of the calculus, the basis of a definition 
of infinite assemblages. 

Although Bolzano's ideas indicate the direction in which the final 
formulation of the calculus lay and which much of the thought of the 
nineteenth century was to follow, they did not constitute the decisive 
influence in determining this. His work remained largely unnoticed 
until rediscovered by Hermann Hankel more than a half century 
later. 13 Fortunately, however, the mathematician A. L. Cauchy pur- 
sued similar ideas at about the same time and was successful in 
establishing these as basic in the calculus. 

Cauchy rivaled Euler in mathematical productivity, contributing 
some 800 books and articles on almost all branches of the subject. 14 
Among his greatest contributions are the rigorous methods which he 
introduced into the calculus in his three great treatises: the Cours 
d 'analyse de I'Ecole Poly technique (1821), Resume des leqons sur le 
calcul infinitesimal (1823), and Leqons sur le calcul differentiel (1829). 15 
Through these works Cauchy did more than anyone else to impress 
upon the subject the character which it bears at the present time. 

We have seen the notion of a limit develop gradually out of the Greek 
method of exhaustion, until it was expressed by Newton in the 
Principia. It was more definitely invoked by Robins, D'Alembert, 
and L'Huilier as the basic concept of the calculus, and as such was 
included by Lacroix in his textbooks. Throughout this long period, 
however, the limit concept lacked precision of formulation. This 
resulted from the fact that it was based on geometrical intuition. It 
could hardly have been otherwise, inasmuch as during this time the 
ideas of arithmetic and algebra were largely established upon those 
of geometrical magnitude. The calculus was interpreted by its in- 
ventors as an instrument for dealing with relationships between 

13 See Dickstein, "Zur Geschichte der Principien der Infinitesimalrechnung," p. 77; also 
Hankel's article "Grenze," in Ersch und Gruber's Allgemeine Encyklopddie. 

14 See Valson, La Vie et les travaux du Baron Cauchy, reviewed by Boncompagni, p. 57. 
16 The first is found in (Euvres (2), III; the last two in (Euvres (2), IV. 



272 The Rigorous Formulation 

quantities involved in geometrical problems, and as such was largely 
accepted by their successors. Euler and Lagrange, in a sense repre- 
sented exceptions to this rule, for they wished to establish the calculus 
on the formalism of their analytic function concept. However, even 
they rejected the limit idea. Furthermore they unwittingly deferred 
to the preconceptions of geometrical intuition when they uncritically 
inferred that their methods were applicable to all continuous curves. 
Although D'Alembert, L'Huilier, and Lacroix had prepared the 
ground for Cauchy by popularizing the limit idea in their works, this 
conception remained largely geometrical. 

In the work of Cauchy, however, the limit concept became, as it 
had in the thought of Bolzano, clearly and definitely arithmetical 
rather than geometrical. Formerly, when illustrations of the notion 
were desired, the one most likely to be called to mind was that of a 
circle defined as the limit of a polygon. Such an illustration immedi- 
ately served to bring up questions as to the manner in which this is 
to be interpreted. Is it the approach to coincidence of the sides of the 
polygon with the points representing the circle? Does the polygon 
ever become the circle? Are the properties of the polygon and the 
circle the same? It was questions such as these that retarded the 
acceptance of the limit idea, for they were similar to those of Zeno 
in demanding some sort of visualization of the passage from the one 
to the other by which the properties of the first figure merge into 
those of the second. 

Quite recently the limit concept has been loosely interpreted in 
the assertion that whether one calls the circle the limit of a polygon 
as the sides are indefinitely decreased, or whether one looks upon it 
as a polygon with an infinite number of infinitesimal sides, is im- 
material, inasmuch as in either case in the end "the specific differ- 
ence" between the polygon and the circle is destroyed. 16 Such an 
appeal to geometrical intuition is quite irrelevant in the case of the 
limit concept. In giving his definition, in his Cours d* analyse, Cauchy 
divorced the idea from all reference to geometrical figures or magni- 
tudes, saying: "When the successive values attributed to a variable 
approach indefinitely a fixed value so as to end by differing from it 
by as little as one wishes, this last is called the limit of all the others." 17 

» Vivanti, // concetto d'infinitesimo, p. 39. " See (Euvres (2), III, 19; cf. also IV, 13. 



The Rigorous Formulation 273 

This is the most clear-cut definition of the concept which had been 
given up to that time, although later mathematicians were to voice 
objections to this also and to seek to make it still more formal and 
precise. Cauchy's definition appealed to the notions of number, vari- 
able, and function, rather than to intuitions of geometry and dy- 
namics. Consequently, in illustrating the limit concept he said that an 
irrational number is the limit of the various rational fractions which 
furnish more and more approximate values of it. 18 

Upon the basis of this arithmetical definition of limit, Cauchy then 
proceeded to define that elusive term, infinitesimal. Ever since the 
time that Greek mathematical speculation had hit upon the infinitely 
small, this notion had been bound up with geometrical intuition of 
spatial properties and had been regarded as a more or less fixed 
minimum of extension. The concept had not thrived in arithmetic, 
largely because unity was considered the numerical minimum, the 
rational fractions having been treated as ratios of two numbers. 
However, the seventeenth century saw the rapid rise of algebraic 
methods in geometry, so that from the time of Fermat the infinitesimal 
had been the concern of both algebra and geometry. Newton had 
insisted that his method did not involve the consideration of minima 
sensibilia, but his interpretation of the procedure in terms of an 
ultimate ratio, rather than of a limiting number, made his evanescent 
magnitudes appear such. Liebniz had been less definite and con- 
sistent in denying the existence of actual infinitesimals, for he some- 
times considered them as assignable, sometimes as inassignable, and 
occasionally as qualitatively zero. However, the development during 
the eighteenth century of the function concept, with its emphasis on 
the relations between variables, led Cauchy to make the infinitesimal 
nothing more than a variable. "One says that a variable quantity 
becomes infinitely small when its numerical value decreases indefi- 
nitely in such a way as to converge toward the limit zero." 19 An 
infinitesimal was consequently not different from other variables, 
except in the understanding that it is to take on values converging 
toward zero as a limit. 

In order to make the concept of the infinitesimal more useful and 
to take advantage of the operational facility afforded by the Leibnizian 

» (Euvres (2), III, 19, 341. » (Euvres (2), III, 19; IV, 16. 



274 The Rigorous Formulation 

views, Cauchy added definitions of infinitesimals of higher order. 
Newton had restricted himself to infinitesimals, or evanescent quan- 
tities, of the first order, but Leibniz had attempted to define those of 
higher orders also. One of the second order, for example, he defined 
as being to one of first order as the latter is to a given finite constant, 
such as unity. Such a vague definition could not be consistently 
applied, but D'Alembert had sought to correct it by an interpretation 
in terms of limits. He had in a sense recognized that the infinitesimals 
were to be regarded as variables and that orders of infinitely small 
magnitudes were to be defined in terms of ratios of these; but his 
work, as we have seen, lacked the precision of statement necessary 
for its general acceptance. It thus remained for Cauchy to express the 
idea of D'Alembert in the precise symbolism of the limit concept. 
Cauchy defined an infinitesimal y = f(x) to be of order n with 

lim/ y\ _ no lim/ y ' 



respect to an infinitesimal x if *-»o( -^zr e ) = and x _>ol -^+- € I = ± oo 

€ having its classical significance — a positive constant, however small. 20 
Here again one recognizes in Cauchy 's work the dominance of the 
ideas of variable, function, and limit. Newton, Leibniz, and D'Alem- 
bert had not distinguished clearly between independent and depend- 
ent infinitesimals, but Cauchy incorporated this in his definition 
when he spoke of the order of the infinitesimal, y, with respect 
to another, x. Thus the latter is the independent variable which 
may be given any sequence of values tending toward as a limit, 
the corresponding sequence of values of y being determined from 
the functional relationship between y and x. Then the limits of the 

sequences of values of the ratios -^- 6 and -^+~ e are found, and the 

order of the infinitesimal y thus determined. This is substantially 
the same as the definition commonly given in present-day textbooks — 
that y is said to be an infinitesimal of order n with respect to another 

infinitesimal x if nm —„ is a constant different from zero. 

In a similar manner Cauchy made rigorously clear the views on 
orders of infinity which D'Alembert had expressed. Whereas Bolzano, 

»Seeffi»w«(2),IV,281. 



The Rigorous Formulation 275 

thinking in terms of aggregation, had asserted the possibility of an 
actual infinity, Cauchy, emphasizing variability, denied the possi- 
bility of this because of the paradoxes to which such an assumption 
appeared to lead. 21 He admitted only the potential infinite of Aris- 
totle, and with D'Alembert interpreted the infinite as meaning simply 
indefinitely large — a variable, the successive values of which increase 
beyond any given number. 22 Orders of infinity he then defined in a 
manner exactly analogous to that given for infinitesimals. 

Having established the notions of limit, infinitesimal, and infinity, 
Cauchy was able to define the central concept of the calculus — that 
of the derivative. His formulation was precisely that given by Bolzano: 
Let the function be y = f(x); to the variable x give an increment 

/\y fix ~\~ t) — f\X) 

Ax = i\ and form the ratio • — = . The limit of this 

Ax i 

ratio ("when it exists") as i approaches zero he represented by f(x), 
and he called this the derivative of y with respect to xP This is, of 
course, the differential quotient of L'Huilier, clarified by the applica- 
tion of the function concept of Euler and Lagrange. It is made the 
central concept of the differential calculus, and the expression "dif- 
ferential" is then defined in terms of the derivative. The differential 
thus represents simply a convenient auxiliary notion permitting the 
application of the suggestive notation of Leibniz without the con- 
fusion between increments and differentials which this symbolism had 
engendered. Leibniz had considered differentials as the fundamental 
concepts, the differential quotient being defined in terms of these; but 
Cauchy reversed this relationship. Having defined the derivative in 
terms of limits, he then expressed the differential in terms of the 
derivative. If dx is a finite constant quantity h, then the differential 
dy of y = f{x) is defined as f{x)dx. In other words, the differentials 

(dy) . ., 
dy and dx are quantities so chosen that their ratio - — - coincides 

{dx) 

with that of the "derniere raison" or the limit y' = f(x) of the ratio 

Ay 

— -, 24 This is practically the view of D'Alembert and L'Huilier, and 

Ax 

21 See Enriques, Historic Development of Logic, p. 135. 

22 (Euvres (2), HI, 19 ff.; cf. also IV, 16. 

■ (Euvres (2), IV, 22; cf. also pp. 287-89. M (Euvres (2) IV, 27-28; 287-89. 



276 The Rigorous Formulation 

even Leibniz had in a sense anticipated it when he said in 1684 that 
dy was to dx as the ratio of the ordinate to the subtangent. To make 
his work logical, however, Leibniz would have had to define the term 
"subtangent" in terms of limits, thus making the limiting ratio the 
derivative. 

Cauchy, however, gave to the derivative and the differential a 
formal precision which had been lacking in the definitions of his 
predecessors. He was therefore able to give satisfactory definitions of 
differentials of higher order also. The differential dy = f(x)dx is, of 
course, a function of x and dx. Regarding dx as fixed, the function 
f(x)dx will in turn have a derivative f'{x)dx and a differential d 2 y = 
f'(x)dx. 2 In general d n y = f{x)dx n . Cauchy added that because the 
nth derivative is the coefficient by which dx n is to be multiplied to 
give d"y, this derivative is called the differential coefficient. 25 

This statement is not to be understood as indicating that deriva- 
tives of higher order are to be defined in terms of differentials of 
higher order. The reverse is of course the case. Differentials have no 
logical significance independent of that of derivatives. They were 
retained by Cauchy simply as an auxiliary notion offering greater 
operational facility than that afforded by derivatives. This fact has 
led the mathematician Hadamard, in connection with a discussion on 
the subject given in the Mathematical Gazette for the years 1934-36, 
to disparage as meaningless the use of differentials of higher order in 
expositions of the calculus. 26 

Cauchy's definitions of the derivative and the differential are not 
in any real sense new. They indicate rather a clarification, by the 
application of the concepts of function, variable, and limit of a vari- 
able, of others previously given. During the eighteenth century the 
word function had generally designated an expression which could be 
written down simply, in terms of variables and symbols of operations 
commonly employed at the time. The nondifferentiable function of 
Bolzano would not, of course, have been included under such an 
understanding. However, early in the next century the work of J. B. 
J. Fourier showed that quite arbitrary discontinuous curves could be 
represented analytically by means of infinite series of trignometric 

25 ffi«w«(2),IV, 301 ff. 

26 Hadamard, "La Notion de diffSrentiel dans Penseignement." 



The Rigorous Formulation 277 

functions. As a consequence, the attitude toward the function concept 
became broader. 27 

The formalistic view that a function was a simple analytic ex- 
pression gave way to the understanding that it was any relationship 
between variables. With this attitude came also the recognition that 
continuity in a curve did not depend upon its being expressible by 
means of a single equation in continuous functions. This, in turn, led 
to the realization that a new definition of continuity was necessary. 
Cauchy's answer to the need was similar to the unnoticed one of 
Bolzano: the function /(x) is continuous within given limits if between 
these limits an infinitely small increment i in the variable x produces 
always an infinitely small increment, f(x + i) — /(x), in the function 
itself. 28 The expressions infinitely small are here to be understood, as 
elsewhere in Cauchy's work, in terms of the indefinitely small and 
limits: i. e., f(x) is continuous within an interval if the limit of the 
variable /(x) as x approaches a is f(a), for any value of a within this 
interval. In this definition the view of the preceding centuries is 
reversed. Newton (implicitly) and Leibniz (explicitly) based the 
validity of the calculus on the assumption, which Greek thought had 
avoided, that, by a vague sense of continuity, limiting states would 
obey the same laws as the variables approaching them. Cauchy made 
the notion of continuity precisely mathematical and showed that this 
depends upon the limiting idea and not vice versa. Furthermore, its 
essence does not lie in a vague blending or unity or contiguity of 
parts, as intuition seems to imply and as Aristotle had stated, but in 
certain formal particulate arithmetical relationships, elaborated later 
in the theory of sets of points, which in turn led to the definition of 
the continuum. 

With the new notion of continuity came a group of new problems. 
It may have been remarked that throughout the period of indecision — 
from Newton and Leibniz to Lagrange and Lacroix — the discussion 
centered about the concepts of the differential and the derivative, to 
the exclusion of that of the integral. The explanation of this is easily 
found. From the time of the Greeks down to that of Pascal, areas 

27 See Jburdain's two articles, "Note on Fourier's Influence on the Conceptions of 
Mathematics," and "The Origin of Cauchy's Conceptions of a Definite Integral and of 
the Continuity of a Function/' 

28 (Euvres (2), III, 43; IV, 19-20, 278. 



278 The Rigorous Formulation 

had been found by various devices equivalent to summations of 
elements. When properly interpreted in terms of the limit concept, 
these methods represented the counterpart of what is now called the 
definite integral. With Barrow, Newton, and Leibniz, however, the 
remarkable discovery was made that the problem of finding areas 
was simply the inverse of that of determining tangents to curves. 
Inasmuch as convenient algorithms — those of fluxions and of dif- 
ferentials — were developed in connection with the latter class of prob- 
lems, by the mere process of inversion the determinations of quad- 
ratures could be systematized. 

The inverse of the fluxion Newton called the fluent. Leibniz had 
himself defined the integral as a sum of differentials, although he 
recognized it also as the inverse of the differential and had deter- 
mined it in accordance with this fact. These inverses of the fluxion and 
the differential are the equivalents of what is now called the anti- 
derivative, or the primitive, or the indefinite integral, or sometimes 
simply the integral. During the period we have been discussing, this 
aspect of the integral as an inverse prevailed over that of the integral 
as a sum. John Bernoulli, in the formal development of the calculus 
summatorius of Leibniz, gave up the definition of the integral as a sum 
and called it definitely the inverse of the differential. He conceived the 
object of the integral calculus as that of finding, from a given relation 
among the differentials, the relation of the quantities themselves. 29 

Euler used the sum conception to find the approximate values of 
definite integrals, but because he interpreted the differential as zero, 
he rejected the Leibnizian view of integration as a process of summa- 
tion and followed John Bernoulli in defining the integral as the inverse 
of the differential. 30 L'Huilier went so far in emphasizing the integral 
as the inverse of the derivative that he suggested substituting the 
expression "rapport integral" for "somme integrate." 31 Lagrange like- 
wise considered the problem of the integral calculus as that of deter- 
mining from the "fonctions derivees" the original function; 32 and 
Lacroix said that the object of the integral calculus was to determine 
from the differential coefficients the functions from which they were 

29 Die erste Integralrechnung, p. 3; cf. also p. 8; also Opera omnia, III, 387. 

80 Opera omnia, XI, 7. 

31 Exposition Slementaire, p. 32. Cf. p. 144. 82 CEuvres, III, 443. 



The Rigorous Formulation 279 

derived. 53 Bolzano similarly defined the integral as the inverse of the 
derivative. 34 

The result of this tendency in the calculus was that logically the 
definition of the integral during this time rested immediately upon 
that of the differential, and the latter became the basis of discussions 
on the validity of the operations and conceptions of the calculus. 
Developments during the early nineteenth century, however, intro- 
duced new points of view. These led to the reinstatement by Cauchy 
of the notion of the (definite) integral as a limit of a sum, and made . 
necessary two independent definitions of the two fundamental con- 
cepts of the calculus, those of the derivative and the integral. 

,. f(x _f_ fj\ — f(x) 

Inasmuch as the derivative has been defined as , im , 

*->o h 

we see from the definition of continuity given by Bolzano and Cauchy 
that the existence of this implied the continuity of the function at the 
value in question, although the converse is not true. The existence of 
an integral in the eighteenth-century sense, that is, as an anti- 
derivative, is therefore bound up with the question of continuity. 
However, even discontinuous curves apparently have an area, and so 
discontinuous functions may allow of an integral in the Leibnizian 
sense. Cauchy therefore restored the character of the definite integral 
as a sum. For a function y = f(x), continuous in the interval from 
Xo to X, he formed the characteristic sum of the products S H = 
(xi - x )f(xo) + (#2 - xi)f(xi) + . . . + (X — x H _ i )f(x n _ l ). If the 
absolute values of the differences x i + l — x { decrease indefinitely, the 
value of S n will "finally attain a certain limit" 5 which will depend 
uniquely on the form of the function fix) and on the limiting values 
x and X . . . "This limit is called a definite integral." 35 

Cauchy cautioned that the symbol of integration J" employed to 
designate this limit was not to be interpreted as a sum, but rather 
as a limit of a sum of this type. 36 Cauchy then brought out the fact 
that although the two operations are defined independently of each 
other, integration in this sense is the inverse of the process of differ- 

« TraiiS du c ale id, n, 1-2. » Bolzano's Schriften, pp. 83-84. 

K (Euvres (2), IV, 125; cf. also Jourdain, "The Origin of Cauchy's Conceptions," pp. 
664 ft.; "The Theory of Functions with Cauchy and Gauss," p. 193. 
»«<E»w«(2), IV, 126. 



280 The Rigorous Formulation 

entiation. He showed that if f(x) is a continuous function, the function 
defined as the definite integral F(x) = jl f(x)dx has as its derivative 
the function f(x). 37 This was perhaps the first rigorous demon- 
stration of the proposition known as the fundamental theorem of the 
calculus. 38 

Cauchy's definition of the definite integral allows of extension, with 
slight modifications, to functions which have discontinuities within 
the interval of definition. If, for example, the function f{x) is dis- 
continuous at the point X within the interval x to X, the definite 
integral from x to X is defined as the limit, if it exists, of the sum 
Ixo ~ e f(%)dx + Jxo + e f( x )dx, as c becomes indefinitely small. 39 

Discontinuous functions have come to play a significant role in 
mathematics and science, and the view of the integral as a sum has 
been that upon which the theory of integration has largely developed 
since the time of Cauchy. From this view, for example, the integral 
of Lebesgue has developed. 40 The manipulation of infinite series, such 
as those entering in the definition of the definite integral, had gone 
on for well over a century before the time of Cauchy, but the need 
for considering the convergence of these had not been strongly felt 
until about the opening of the nineteenth century. The term "conver- 
gent series" seems first to have been used, although in a somewhat 
restricted sense, a century and a half earlier by James Gregory; 41 
but the general lack of rigor in the work of the eighteenth century 
was uncongenial to the precision of thought necessary to develop this 
idea. Varignon, at the beginning of the century, and Lagrange, at its 
close, had gone a step in this direction by saying that no series could 
safely be used unless one investigated the remainder. 42 

Nevertheless, no general definition, or theory, of convergent infinite 
series had been given, and Euler and Lacroix continued to employ 
divergent series in their work. With the turn of the next century, 
however, Abel, Bolzano, Cauchy, and Gauss all pointed out the 
need for definitions and tests of convergence of infinite series before 
the latter could legitimately be employed in mathematics. In this 

* Ibid., pp. 151-52. M Saks, Tktorie de Vintigrale, pp. 122-23. 

39 See Cauchy, (Euvres (2), I, 335 and (Euvres (1), I, 335. 

40 Saks, Thiorie de I'intigrale, p. 125. 

41 See Vera circuli et hyperbolae quadratura, p. 10. 

42 Reiff, Geschichte der unendlichen Reihen, pp. 69-70, 155. 



The Rigorous Formulation 281 

respect the work of Cauchy, in particular, laid the foundation of the 
theory of convergence and divergence through the wide influence which 
his work exerted upon his contemporaries. Cauchy defined a series as 
convergent if, for increasing values of n, the sum S n approaches indefi- \ 
nitely a certain limit S, the limit S in this case being called the sum 
of the series. 43 Cauchy here showed clearly that the limit notion is 
involved, as it was also in differentiation and integration and in 
defining continuity. Furthermore, he pointed out that it is only in i 
this sense that an infinite series may be regarded as having a sum. ; 
In other words, Zeno's paradox of the Achilles is to be answered in 
precisely such ideas, based upon the limit concept. 

Cauchy went on in this work to try to prove what has become 
known as Cauchy's theorem — that a necessary and sufficient condi- 
tion that the sequence converge to a limit is that the difference be- 
tween S P and S q for any values of p and q greater than n can be 
made less in absolute value than any assignable quantity by taking 
n sufficiently large. A sequence satisfying this condition is now said 
to converge within itself. The necessity of the condition follows im- 
mediately from the definition of convergence, but the proof of the 
sufficiency of the condition requires a previous definition of the system 
of real numbers, of which the supposed limit 5 is one. Without a 
definition of irrational numbers, this part of the proof is logically im- 
possible. 

Cauchy had stated in his Cours d 'analyse that irrational numbers 
are to be regarded as the limits of sequences of rational numbers. 
Since a limit is defined as a number to which the terms of the sequence 
approach in such a way that ultimately the difference between this 
number and the terms of the sequence can be made less than any 
given number, the existence of the irrational number depends, in the 
definition of limit, upon the known existence, and hence the prior 
definition, of the very quantity whose definition is being attempted. 
That is, one cannot define the number V 2 as the limit of the sequence 
1, 1.4, 1.41, 1.414, . . . because to prove that this sequence has a limit 
one must assume, in view of the definitions of limit and convergence, 
the existence of this number as previously demonstrated or defined. 

Cauchy appears not to have noticed the circularity of the reasoning 

« (Euvres (2), HI, 114. 



282 The Rigorous Formulation 

in this connection, 44 but tacitly assumed that every sequence con- 
verging within itself has a limit. That is, he felt that the existence of a 
number possessing the external relationship expressed in the definition 
of convergence and the sum of the series, would follow from the inner 
relations expressed in the Cauchy theorem. This idea may have been 
based upon the very thing that he and Bolzano had sought to avoid — 
that is, upon preconceptions taken over from geometry. The attempt 
to base the idea of number upon that of the geometrical line had given 
rise to the Pythagorean difficulty of the incommensurable and the 
ensuing development of the calculus. It had likewise suggested to 
Gregory of St. Vincent, two centuries before Cauchy, that the sum 
of an infinite geometrical progression could be represented by the 
length of a line segment and that the series could therefore be thought 
of as having a limit. However, in order to make the limit concept of 
analysis independent of geometry, mathematicians of the second half 
of the nineteenth century attempted to frame definitions of irrational 
number which did not make use of the definition of a limit. 

The geometrical intuitions which intruded themselves into Cauchy's 
view of irrational number likewise led him erroneously to believe that 
the continuity of a function was sufficient for its geometrical repre- 
sentation and for the existence of a derivative. 46 A. M. Ampere also 
had been led by geometric preconceptions similar to those of Cauchy 
to try to demonstrate the false proposition that every continuous 
function has a derivative, except for certain isolated values in the 
interval. 46 Bolzano had in his manuscripts of about this time given an 
example showing the falsity of such an opinion, but it remained for 
Weierstrass to make this fact known. 

With Cauchy, it may safely be said, the fundamental concepts of 
the calculus received a rigorous formulation. Cauchy has for this 
reason commonly been regarded as the founder of the exact differential 
calculus in the modern sense. 47 Upon a precise definition of the notion 
of limit, he built the theory of continuity and infinite series, of the 
derivative, the differential, and the integral. Through the popularity 
of his lectures and textbooks, his exposition of the calculus became 

44 Cf. Pringsheim, "Nombres irrationnels et notion de limite," p. 180. 

45 Cf. Jourdain, "The Theory of Functions with Cauchy and Gauss.'' 

48 See Pringsheim, "Principes fondamentaux de la theorie des fonctions." 
47 Klein, Elementary Mathematics from an Advanced Standpoint, p. 213. 



The Rigorous Formulation 283 

that generally adopted and the one which has been accepted down to 
the present time. Nevertheless, the use of the infinitely small per- 
sisted for some time. S. D. Poisson, in his Traite de mecanique which 
appeared in several editions in the first half of the nineteenth century 
and which was long a standard work, used exclusively the method of 
infinitesimals. These magnitudes, "less than any given magnitude of 
the same nature," he held to have a real existence. They were not 
simply "a means of investigation imagined by geometers." 48 The 
object of the differential calculus he consequently regarded as the 
determination of the ratio of infinitely small quantities, in which 
infinitesimals of higher order were neglected; 49 and the integral was 
the inverse of the differential quotient. 

A. A. Cournot likewise opposed the work of Cauchy, although upon 
somewhat different grounds. In his Traite elementaire de la theorie des 
jonctions et du calcul infinitesimal of 1841, he asserted that his taste 
for the philosophy of science prepared him to treat the metaphysics 
of the calculus. 50 In presenting this, his attitude resembled some- 
what that of Carnot. He held that the theories of Newton and Leibniz 
complemented each other, and that the method of Lagrange repre- 
sented simply a return to the views of Newton. 51 The infinitely small 
"existed in nature" as a mode of generation of magnitudes according 
to the law of continuity, although it could be defined only indirectly 
in terms of limits. 52 

However, Cournot protested that concepts exist in the under- 
standing, independently of the definition which one gives to them. 
Simple ideas sometimes have complicated definitions, or even none. 
For this reason he felt that one should not subordinate the precision 
of such ideas as those of speed or the infinitely small to logical defi- 
nition. 53 This point of view is diametrically opposed to that which 
has dominated the mathematics of the last century. The tendency in 
analysis since the time of Cournot has been toward ever-greater care 
in the formal logical elaboration of the subject. This trend, initiated 
in the first half of the nineteenth century and fostered largely by 
Cauchy, was in the second half of that century continued with notable 
success by Weierstrass. 

* Traits de mfcanique, I, 13-14. * Ibid., pp. 14-16. 

60 Traite elementaire, Preface. 61 Ibid. B2 Ibid., pp. 85-88. M Ibid., p. 72. 



284 The Rigorous Formulation 

In spite of the care with which Cauchy worked, there were a num- 
ber of phrases in his exposition which required further explanation. 
The expressions "approach indefinitely," "as little as one wishes," 
"last ratios of infinitely small increments," were to be understood in 
terms of the method of limits, but they suggested difficulties which 
had been raised in the preceding century. The very idea of a variable 
approaching a limit called forth vague intuitions of motion and the 
generation of quantities. Furthermore, there were, in Cauchy's pres- 
entation, certain subtle logical gaps. One of these was the failure to 
make clear the notion of an infinite aggregate, which is basic in his 
work in infinite sequences, upon which the derivative and the integral 
are built. Another lacuna is evident in his omission of a clear defi- 
nition of that most fundamental of all notions — number — which is 
absolutely essential to the definition of limits, and therefore to that 
of the concepts of the calculus. The first of these points had been 
touched upon by Bolzano, but the theory was not further developed 
until much later, largely through the efforts of Georg Cantor. In the 
second matter the difficulty is essentially that of a vicious circle in 
the definition of irrational numbers, and this Weierstrass sought to 
resolve. 

Although it was Cauchy who gave to the concepts of the calculus 
their present general form, based upon the limit concept, the last 
word on rigor had not been said, for it was Karl Weierstrass 54 who 
constructed a purely formal arithmetic basis for analysis, quite 
independent of all geometric intuition. Weierstrass in 1872 read a 
paper in which he showed what had been known to Bolzano sometime 
before — that a function which is continuous throughout an interval 
need not have a derivative at any point in this interval. 55 Previously 
it had been generally held, upon the basis of physical experience, that 
a continuous curve necessarily possessed a tangent, except perhaps at 
certain isolated points. From this it would follow that the correspond- 
ing function should in general possess a derivative. Weierstrass, how- 
ever, demonstrated conclusively the incorrectness of such suggestions 

w See Poincare, "L'CEuvre mathematique de Weierstrass"; cf. also Pierpont, "Mathe- 
matical Rigor," pp. 34r-36. 

65 This seems to have been presented by Weierstrass in his lectures as early as 1861. 
See Pringsheim, "Principes fondamentaux," p. 45, n.: Voss, "Calcul differentiel," pp. 
260-61. ™ 



The Rigorous Formulation 285 

of experience. This he did by forming the nondifferentiable contin- 

OS 

uous function f(x) = 2 b n cos{a n Tx), where x is a real variable, a an 
» = o 

odd integer, and b a positive constant less than unity such that 

ab > 1 + — .«* 

2 

Since that time many other such functions have become known, and 
we may even say that, in spite of geometric intuition, of all contin- 
uous functions those with tangents at some points are the exceptions." 
Intuition has been even more discredited as a guide by the fact that 
we can have continuous curves defined by motion, which yet have 
no tangents. 58 

Inasmuch as it was apparent to Weierstrass that intuition could not 
be trusted, he sought to make the bases of his analysis as rigorously 
and precisely formal as possible. He did not present his work on the 
elements of the calculus in a number of treatises, as had Cauchy, nor 
even in a series of papers. His views became known, rather, through 
the work of students who attended his lectures. 69 

In order to secure logical exactitude, Weierstrass wished to estab- 
lish the calculus (and the theory of functions) upon the concept of 
number alone, 60 thus separating it completely from geometry. To do 
this it was necessary to give a definition of irrational number which 
should be independent of the limit idea, since the latter presupposes 
the former. Weierstrass was thus led to make profound investigations 
into the principles of arithmetic, particularly with respect to the theory 
of irrationals. In this work Weierstrass did not go into the nature 
of the whole number itself, but began with the concept of whole 
number as an aggregate of units enjoying one characteristic property 
in common, whereas a complex number was to be thought of as an 
aggregate of units of various species enjoying more than one char- 

56 See Weierstrass, Mathematische Werke, II, 71-74; Mansion, "Fonction continue sans 
derivee de Weierstrass." 

67 Cf. Voss, "Calcul differentiel," pp. 261-62. 

M Neikirk, "A Class of Continuous Curves Defined by Motion Which Have No Tan- 
gents Lines." 

69 See, for example, Pincherle, "Saggio di una introduzione alia teoria delle funzioni 
analitiche secondo i principii del Prof. C. Weierstrass." Cf. Merz, A History of European 
Thought in the Nineteenth Century, II, 703. 

60 Jourdain, "The Development of the Theory of Transfinite Numbers," 1908-9, n., p. 
298; cf. also p. 303. 



286 The Rigorous Formulation 

acteristic property. All rational numbers can then be denned by 
introducing convenient classes of complex numbers. Thus the number 
3f is made up of 3 a and 2/3, where a is the principle unit and /3 is an 
aliquot part, §, taken as another element. A number is then said to. 
be determined when we know of what elements (of which there is an 
infinite number) it is composed and the number of times each occurs. 
In this theory the number V 2 is not denned as the limit of the sequence 
1, 1.4, 1.41, ... , nor is the idea of sequence brought in; it is simply 
the aggregate itself in any order l a , 40, I7, . . . where a is the prin- 
ciple unit and p, y, . . . are certain of its aliquot parts, and where the 
aggregate is, of course, subject to the condition that the sum of any 
finite number of elements is always less than a certain rational num- 
ber. We can now prove, if we wish, that this number is the limit of 
the variable sequence l a ; la, 40; U, 4/3, 1 T ; . . . , thus correcting the 
logical error arising in Cauchy's theory of number and limits. 61 In a 
sense, Weierstrass settles the question of the existence of a limit of a 
convergent sequence by making the sequence (really he considers an 
unordered aggregate) itself the number or limit. 

In making the basis of the calculus more rigorously formal, Weier- 
strass also attacked the appeal to intuition of continuous motion which 
is implied in Cauchy's expression — that a variable approaches a limit. 
Previous writers generally had denned a variable as a quantity or 
magnitude which is not constant; but since the time of Weierstrass it 
has been recognized that the ideas of variable and limit are not 
essentially phoronomic, but involve purely static considerations. 
Weierstrass interpreted a variable * as simply a letter designating any 
one of a collection of numerical values. 62 A continuous variable was 
likewise defined in terms of static considerations: If for any value 
x of the set and for any sequence of positive numbers 5 i, 5 2 , . • • , 
8„, however small, there are in the intervals x — g,-, x + 8{ others of 
the set, this is called continuous. 63 

Similarly, for a continuous function Weierstrass gave a definition 
equivalent to those of Bolzano and Cauchy, but having greater clarity 
and precision. To say that f(x + A*) - f(x) becomes infinitesimal, 

61 Ibid., 1908-9, pp. 303 ff. Cf. Russell, Principles of Mathematics, pp. 281 ff.; Pring- 
sheim, "Nombres irrationnels et notion de limite," pp. 149 ff.; Pincherle, op. cit., pp. 179 ff. 
« Pincherle, op. cit., p. 234. « Ibid., p. 236. 



The Rigorous Formulation 287 

or becomes and remains less than any given quantity, as A* ap- 
proaches zero, calls to mind either the infinitely small or else vague 
notions of mobility. Weierstrass defined f{x) as continuous, within 
certain limits of x, if for any value x in this interval and for an arbi- 
trarily small positive number e, it is possible to find an interval about 
x such that for all values in this interval the difference f(x) — f(x ) 
is in absolute value less than e ; 64 or, as Heine was led by the lectures 
of Weierstrass to express it, if, given any e , an v can be found such 
that for V < \, the difference fix ± v) — f(x) is less in absolute 
value than e. 65 

The limit of a variable or function is similarly defined. The number 
L is the limit of the function f(x) for x = x if, given any arbitrarily 
small number c, another number 5 can be found such that for all 
values of x differing from x by less than 5, the value of f(x) will 
differ from that of L by less than e. 66 This expression of the limit idea, 
in conjunction with Cauchy's definitions of the derivative and the 
integral, supplied the fundamental conceptions of the calculus with a 
precision which may be regarded as constituting their rigorous formula- 
tion. There is in this definition no reference to infinitesimals, so that 
the designation "the infinitesimal calculus," which is used even today, 
is shown to be inappropriate. Although a number of mathematicians, 
from the time of Newton and Leibniz to that of Bolzano and Cauchy, 
had sought to avoid the use of infinitely small quantities, the un- 
equivocal symbolism of Weierstrass may be regarded as effectively 
banishing from the calculus the persistent notion of the fixed infin- 
itesimal. 

During the eighteenth century there had been a lively argument, 
both in connection with the prime and ultimate ratio of Newton 
and with the differential quotient of Leibniz, as to whether a variable 
which approaches its limit can ever attain it. This is essentially the 
crux of Zeno's argument in the Achilles. In the light of the precision 
of the Weierstrassian theory of limits, however, the question is seen to 
be entirely inapposite. The limit concept does not involve the idea of 
approaching, but only a static state of affairs. The single question 

M Ibid., p. 246. 

65 Heine, "Die Elemente der Funktionenlehre," p. 182; cf . also p. 186. 
M See Stolz, Vorlesungen fiber allgemeine Arithmetik, I, 156-57; cf. also Whitehead, An 
Introduction to Mathematics, pp. 226-29. 



288 The Rigorous Formulation 

amounts really to two : first, does the variable f(x) have a limit L 
for the value a of x. Secondly, is this limit L the value of the function 
for the value a of x. If f(a) = L, then one can say that the limit 
of the variable for the value of x in question is the value of the 
variable for this value of x, but not that/(V) reaches j '(a) or L, for this 
latter statement has no meaning. 

In retrospect, it is pertinent to remark that whereas the idea of 
variability had been banned from Greek mathematics because it led 
to Zeno's paradoxes, it was precisely this concept which, revived in 
the later Middle Ages and represented geometrically, led in the 
seventeenth century to the calculus. Nevertheless, as the culmination 
of almost two centuries of discussion as to the basis of the new anal- 
ysis, the very aspect which had led to its rise was in a sense again 
excluded from mathematics with the so-called "static" theory of the 
variable which Weierstrass had developed. The variable does not 
represent a progressive passage through all the values of an interval, 
but the disjunctive assumption of any one of the values in the inter- 
val. Our vague intuition of motion, although remarkably fruitful in 
having suggested the investigations which produced the calculus, was 
found, in the light of further elaboration in thought, to be quite 
inadequate and misleading. What, then, about that obscure and 
elusive feeling for continuity which colors so much of our thought? 
Is that baseless also? What about the idea of infinity upon which 
Bolzano had speculated and which Weierstrass had used, somewhat 
covertly, in his definition of irrational number? Can this be given a 
consistent definition? These questions were investigated largely by 
Dedekind and Cantor, two mathematicians who were thinking along 
lines similar to those which Weierstrass had followed in seeking a sat- 
isfactory definition of irrational numbers. 

The year 1872 was, for a number of reasons, a significant one in the 
history of the foundations of the calculus. It saw, besides the pres- 
entation by Weierstrass of his continuous nondifferentiable function 
and the publication by one of his students of Weierstrass' lectures on 
the elements of arithmetic, 67 the appearance of the following: Nouveau 
precis d' analyse infinitesimale of Charles Meray; a paper in Crelle's 
Journal by Eduard Heine on "Die Elemente der Funktionenlehre" ; 

67 Kossak, Die Elemente der Arithmetik. 



The Rigorous Formulation 289 

the first paper by Georg Cantor on the principles of arithmetic, which 
appeared in the Mathematische Annalen as "Uber die Ausdehnung 
eines Satzes aus der Theorie der trigonometrischen Reihen"; and the 
Stetigkeit und die Irrationalzahlen of Richard Dedekind. Incidentally 
the work of each of these men touched upon one and the same prob- 
lem — that of formulating a definition of irrational number which 
should be independent of that of the limit concept. 68 The work of 
Weierstrass in this connection has already been described. With 
respect to publication, this had been anticipated by Meray, who in 
1869, in an article entitled "Remarques sur la nature des quantites 
definies par la condition de servir de limites a des variables donnees," 
sought to resolve the vicious circle in the definitions of limit and 
irrational number given by Cauchy. Three years later Meray further 
elaborated his views, in his Nouveau precis. 

It will be recalled that Bolzano and Cauchy had attempted to 
prove that a sequence which converges within itself — that is, one for 
which, given any e, however small, an integer N can be found such 
that for n > N and for any integral value of p greater than the integer 
n, the inequality \S n + P - Sj, < e will hold— converges in the sense of 
external relations, that is, that it has a limit S. Meray, in this con- 
nection, cut the Gordian knot by rejecting Cauchy's definition of 
convergence in terms of the limit S. He called an infinite series con- 
vergent if it converged within itself, according to Cauchy's theorem. 
In this case one need not demonstrate the existence of an undefined 
number S, which may be regarded as the limit. The word number in 
the strict sense Meray reserved for the integers and rational fractions; 
the converging sequence of rational numbers, which Meray called a 
convergent "variante," he regarded as determining a number in the 
broad sense, rational or irrational. He was somewhat vague as to 
whether or not the sequence is the number. 69 If so, as is implied in the 
case of irrationals, his theory is equivalent to that of Weierstrass, 
although somewhat less explicitly expressed. 

Attempts, similar to those of Weierstrass and Meray, to avoid the 

88 For a full account of this work, together with bibliographical references, see Jourdain, 
"The Development of the Theory of Transfinite Numbers"; see also Pringsheim, "Nombres 
irrationnels et notion de limite," pp. 144 ff. 

69 See "Remarques sur la nature des quantites"; also Nouveau pricis, pp. xv, 1-7; cf. 
also Jourdain, "The Development of the Theory of Transfinite Numbers,'' 1910, pp. 28 ff . 



290 The Rigorous Formulation 

petitio principii in Cauchy's reasoning on limits and irrational num- 
bers were developed and published, also in 1872, by Cantor and by 
Heine. Meray had avoided the logical difficulty by taking \S n + p — 
S n \ < e as the definition of convergence, instead of the condition 
\S — S n \ < e, which presupposes the demonstrated existence of S. 
Likewise in Weierstrass' definition the irrational numbers are ex- 
pressed in an analogous manner, not as limits but as entire infinite 
groups of rational numbers. The work of these men led Heine and 
Cantor to express similar views. Rather than postulate the existence 
of a number S, which is the limit of an infinite series which converges 
within itself, they considered S, not exactly as determined by the 
series, as Meray had somewhat indecisively held, but as defined by 
the series — as simply a symbol for the series itself. 70 Their definitions 
resemble the view of Weierstrass, with the addition of the condition 

of Meray that ^ (S n + P - S H ) = for p arbitrary. This condition is 

equivalent to that of Weierstrass — whose aggregates were such that 
however summed in finite number, the sum was to remain below a 
certain limit — but is expressed in a rather more convenient form. 

Still another attempt along these lines was made by Dedekind. 
Weierstrass had been lecturing on the theory of functions in 1859 
and had in this way been led to investigations concerning the founda- 
tions of arithmetic. In like fashion Dedekind admitted that his atten- 
tion was directed to these matters when in 1858 he found himself 
obliged to lecture for the first time on the elements of the differential 
calculus. 71 In discussing the notion of the approach of a variable 
magnitude to a fixed limiting value, he had recourse, as had Cauchy 
before him, to the evidence of the geometry of continuous magnitude. 
He felt, however, that the theory of irrational numbers, which lay at 
the root of the difficulty in the limit concept, should be developed 
out of arithmetic alone, if it were to be rigorous. 72 

Dedekind 's approach to the problem was somewhat different from 
that of Weierstrass, Meray, Heine, and Cantor in that, instead of 

70 Heine, "Die Elemente der Funktionenlehre," pp. 174 ff.; Cantor, Gesammelte Abltand- 
lungen, pp. 92-102, 185-86. See also the articles by Jourdain, "The Development of the 
Theory of Transfinite Numbers," 1910, pp. 21-43, and "The Introduction of Irrational 
Numbers." Cf. Russell, The Principles of Mathematics, pp. 283-86. 

71 Dedekind, Essays on the Tlieory of Numbers, p. 1. 
» Ibid., pp. 1-3; cf. also p. 10. 



The Rigorous Formulation 291 

considering in what manner the irrationals are to be defined so as to 
avoid the vicious circle of Cauchy, he asked himself what there is in 
continuous geometrical magnitude which resolved the difficulty when 
arithmetic apparently had failed: i. e., what is the nature of con- 
tinuity? Plato had sought to find this in a vague flowing of magni- 
tudes; Aristotle had felt that it lay in the fact that the extremities of 
two successive parts were coincident. Galileo had suggested that it 
was the result of an actually infinite subdivision — that the continuity 
of a fluid was in this respect to be contrasted with the finite, dis- 
continuous subdivision illustrated by a fine powder. The philosophy 
and mathematics of Leibniz had led him to agree with Galileo that 
continuity was a property concerning disjunctive aggregation, rather 
than a unity or coincidence of parts. Leibniz had regarded a set as 
forming a continuum if between any two elements there was always 
another element of the set. 73 

The scientist Ernst Mach likewise regarded this property of the 
denseness of an assemblage as constituting its continuity, 74 but the 
study of the real number system brought oat the inadequacy of this 
condition. The rational numbers, for example, possess the property of 
denseness and yet do not constitute a continuum. Dedekind, thinking 
along these lines, found the essence of the continuity of a line to be 
brought out, not by a vague hang-togetherness, but in the nature of 
the division of the line by a point. He saw that in any division of the 
points of a line into two classes such that every point of the one is to 
the left of every point of the other, there is one and only one point 
which produces this division. This is not true of the ordered system of 
rational numbers. This, then, was why the points of a line formed a 
continuum, but the rational numbers did not. As Dedekind expressed 
it, "By this commonplace remark the secret of continuity is to be 
revealed." 75 

It is obvious, then, in what way the domain of rational numbers is 
to be rendered complete to form a continuous domain. It is only neces- 
sary to assume the Cantor-Dedekind axiom that the points of a line 
can be put into one-to-one correspondence with the real numbers. 
Arithmetically expressed, this means that for every division of the 

73 Pkilosophische Schriften, II, 515. 

74 Die Principien der Warmelehre, p. 71. 76 Essays on the Theory of Numbers, p. 11. 



292 The Rigorous Formulation 

rational numbers into two classes such that every number of the first, 
A, is less than every number of the second, B, there is one and 
only one real number producing this Schnitt, or "Dedekind Cut." 
Thus if we divide the rational numbers into two classes A and B, 
such that A contains all those whose squares are less than two and 
B all those whose squares are more than two, there is, by this axiom 
of continuity, a single real number — written in this case as V 2 — which 
produces this division. Furthermore, this cut constitutes the definition 
of the number V 2. Similarly, any real number is defined by such 
a cut in the rational number system. This postulate makes the domain 
of real numbers continuous, in the sense that the straight line has this 
property. Moreover, the real number of Dedekind is in a sense a crea- 
tion of the human mind, independent of intuitions of space and time. 

The calculus had been generally recognized as dealing with contin- 
uous magnitude, but before this time no one had explained precisely 
the sense in which this was to be accepted. Symbols for variables 
had displaced the idea of geometrical magnitude, but Cauchy im- 
plied a geometrical interpretation of a continuous variable. Dedekind 
showed that it was not, as had frequently been held, 76 the apparent 
freedom from the discreteness of the rational numbers which made 
geometrical quantities continuous, but only the fact that the points 
in them formed a dense, perfect set. On completing the number system 
in the manner suggested by this fact — that is, by adopting Dedekind's 
postulate — this system was made continuous also. Now the funda- 
mental theorems on limits could be proved rigorously 77 and without 
recourse to geometry, as Dedekind pointed out, on the basis of his 
new definition of real number. 78 Geometry having pointed the way 
to a suitable definition of continuity, it was in the end excluded from 
the formal arithmetical definition of this concept. 

The Dedekind Cut is in a sense equivalent to the definitions of real 
number given by Weierstrass, Meray, Heine, and Cantor. 79 Bertrand 

76 See Drobisch, "Ueber den Begriff des Stetigen und seine Beziehungen zum Calcul," 
p. 170. 

77 It should be noted, however, that Dedekind's definition of number has recently been 
criticized as involving a vicious circle. See Weyl, "Der Circulus vitiosus in der heutigen 
Begriindung der Analysis." 

78 Essays on the Theory of Numbers, p. 27; cf. also pp. 35-36. 

79 See J. Tannery, Review of Dantscher, Vorlesungen iiber die Weierstrassche Theorie der 
irrationalen Zahlen. 



The Rigorous Formulation 293 

Russell followed the line of thought suggested by these men, in at- 
tempting another formal definition of real number. He felt that the 
definitions previously given either disregarded the question of the 
existence of the irrational numbers or artificially postulated new 
numbers, leaving some doubt as to just what they are. He suggested 
that a real number be defined as a whole "segment" of the rational 
numbers. The number V2, for example, is defined as the ordered 
aggregate of all rational numbers whose squares are less than two. 
That is, instead of postulating an element dividing the rational num- 
bers into two classes, as Dedekind did, he would merely take one of 
Dedekind's classes and make it, rather than the cutting element, the 
number. 80 This obviates the necessity of introducing any conception 
other than that of rational number and segment of rational numbers. 
According to this view, there is no need to create the irrational num- 
bers; they are at hand in the system of rational numbers, as they had 
been also in the somewhat more involved doctrine of Weierstrass. 

The object of all the above efforts in the establishment of the real 
number was to give a formal logical definition which should be inde- 
pendent of the implications of geometry and which should avoid the 
logical error of defining irrational numbers in terms of limits, and 
vice versa. From these definitions, then, the basic theorems on limits 
in the calculus can be derived without circularity in reasoning. The 
derivative and the integral are thus established directly on these 
definitions, and are consequently divested of any character connected 
with sensory perception, such as rate of change or surface area. Geo- 
metrical conceptions cannot be made sufficiently explicit and precise, 
as we have seen during our consideration of the long period of devel- 
opment of the calculus. Thus the required rigor was found in the 
application of the concept of number, made formal by divorcing it 
from the idea of geometrical quantity. From the definitions of number 
given above, we see that it is not magnitude which is basic, but order. 
This is brought out most clearly in the definitions given by Dedekind 
and Russell, these involving only ordered classes of elements. The 
same is true, however, of the other systems, which have the dis- 
advantage of requiring new definitions of equality before this can be 
made clear. The essential characteristic of the number two is not its 

80 Russell, Introduction to Mathematical Philosophy, p. 72. 



294 The Rigorous Formulation 

magnitude, but its place in the ordered aggregate of real numbers. 
The derivative and the integral, although still defined as limits of 
characteristic quotients and sums respectively, have, as a result, ulti- 
mately become, through the definition of number and limit, not 
quantitative but ordinal concepts. The calculus is not a branch of the 
science of quantity, but of the logic of relations. 

Dedekind's work not only met the need for a definition of number 
independent of that of limit, but in addition gave an explanation of 
the nature of continuous magnitude. Bolzano, Cauchy, and others 
had given definitions of a continuous function of an independent 
variable. A continuous, independent variable was tacitly understood 
as one which could take on all values in an interval corresponding to 
the points of a line segment. The arithmetization of 1872, however, 
went beyond the geometrical picture and expressed formally, in terms 
of ordered aggregates, what was meant by a continuous variable or 
ensemble. The conditions were: first, that the values or elements 
should form an ordered set; second, that this should be a dense set — 
that is, between any two values or elements, there should always be 
others; and third, that the set should be perfect — that is, if the ele- 
ments are divided, as in a Dedekind Cut, there should always be one 
which produces this cut. 

This definition is far removed from any appeal to empiricism and 
from the picture of a smooth, unbroken "oneness" or cohesiveness, 
which instinctive feeling associates with the notion of continuity. It 
specifies only an infinite, discrete multiplicity of elements, satisfying 
certain conditions — that the set be ordered, dense, and perfect. This 
is the sense in which one is to interpret the remark that the calculus 
deals with continuous variables; the sense in which one is to interpret 
Newton's phrase "prime and ultimate ratios," or the ultimate rela- 
tionship between the differentials which Leibniz thought subsisted by 
virtue of the law of continuity. The introduction of uniform motion 
into Newton's method of fluxions was an irrelevant evasion of the 
question of continuity, disguised by an appeal to intuition. There is 
nothing dynamic in the idea of continuity, nor, so far as we know, is 
the converse necessarily true. By sense perception we are apparently 
unable to conclude whether or not we are dealing in motion with a 
continuum. The experiments of Helmholtz, Mach, and others have 



The Rigorous Formulation 295 

shown that the physiological spaces of touch and sight are themselves 
discontinuous. 81 

The continuity of time which Barrow and Newton regarded as 
assured by its relentless even flow is now seen to be simply a hy- 
pothesis. Mathematics is unable to specify whether motion is con- 
tinuous, for it deals merely with hypothetical relations and can make 
its variable continuous or discontinuous at will. The paradoxes of 
Zeno are consequences of the failure to appreciate this fact and of 
the resulting lack of a precise specification of the problem. The dy- 
namic intuition of motion is confused with the static concept of con- 
tinuity. The former is a matter of scientific description a posteriori, 
whereas the latter is a matter solely of mathematical definition a 
priori. The former may consequently suggest that motion be defined 
mathematically in terms of continuous variables, but cannot, because 
of the limitations of sensory perception, prove that it must be so 
defined. If the paradoxes of Zeno are thus stated in the precise mathe- 
matical terminology of continuous variables and of the derived con- 
cepts of limit, derivative, and integral, the seeming contradictions 
resolve themselves. The dichotomy and the Achilles depend upon the 
question as to whether or not the sets involved are perfect. 82 The 
stade is answered upon the basis of dense sets, and the arrow by the 
definition of instantaneous velocity, or the derivative. 

The mathematical theory of continuity is based, not on intuition, 
but on the logically developed theories of number and sets of points. 
The latter, however, depend, in turn, upon the idea of an infinite 
aggregate, an idea which Zeno had invoked to fortify his arguments. 
Zeno's appeal to the infinite was based upon the supposed incon- 
ceivability of the notion of completing in a finite time an infinite 
number of steps. It is again the scientific description a posteriori 
which he questioned, but, so far as we know, there is no way of proving 
or disproving the possibility, not only of the existence of infinite 
aggregates in the physical sense, but also of the execution in thought 
of an infinite number of steps in connection with aggregates, whether 
finite or infinite. Since science cannot answer this point, the question 
may become a hypothetical mathematical one. 

81 Enriques, Problems of Science, pp. 211-12. 

82 Cf. Helmholtz, Counting and Measuring, p. xviii; Cajori, "History of Zeno's Argu- 
ments on Motion," p. 218. 



296 The Rigorous Formulation 

Mathematics, moreover, requires a theory of the infinite in its 
definitions of number and continuity. The question of the mathe- 
matical existence, i. e., of the consistent logical definition, of infinite 
aggregates therefore remained to be answered. Galileo had suggested 
vaguely, and Bolzano had seen more clearly, that infinite ensembles 
must have the paradoxical property that a part can be put into one- 
to-one correspondence with the whole. This fact led Cauchy to deny 
their existence; and indeed upon the basis of the work of Cauchy and 
Weierstrass, one could have said that the infinite indicated nothing 
more than the potentiality of Aristotle — an incompleteness of the 
process in question. 83 Their infinitesimals were variables having zero 
as their limit, and the limit concept involved only the definition of 
number. However, this very definition of number implicitly pre- 
supposes the prior existence of infinite aggregates, so that this ques- 
tion could not indefinitely be avoided. Under the influence of Weier- 
strass' work in the foundations of arithmetic, Dedekind and Cantor 
sought a basis for the theory of infinite aggregates, 84 in order to com- 
plete this work. This they found in Bolzano's paradox. Instead of 
looking upon it as merely a strange property of infinite aggregates, 
they made it the definition of an infinite set. Dedekind said "A system 
5 is said to be infinite when it is similar to a proper part of itself; in 
the contrary case 5 is said to be a finite system." 85 Under this defi- 
nition, infinite aggregates exist as logically self-consistent entities, and 
the definitions of real number are completed. 

Cantor, with whom Dedekind corresponded in this connection, 86 
was not satisfied with merely defining infinite sets. He wished to de- 
velop the subject further. In a series of papers he reviewed the history 
of the infinite from the time of Democritus to that of Dedekind, and 
elaborated his theory of infinite ensembles, or Mengenlehre. Cantor's 
doctrine of the mathematical infinite, which has been hyperbolized as 
"the only genuine mathematics since the Greeks," 87 did not refer to 
the potential infinity of Aristotle nor the syncategorematic infinity of 
the Scholastics. These were bound up always with variability. 88 
Cantor referred instead to the categorematic infinity of medieval 

83 See Baumann, "Dedekind und Bolzano." M See Hilbert, "Uber das Unendliche." 

85 Dedekind, Essays on the Theory of Numbers, p. 63. 

86 See Georg Cantor, Gesammelte Abhandlungen. 

87 See Bell, The Queen of the Sciences, p. 104. w Ibid., p. 180. 



The Rigorous Formulation 297 

philosophy — the eigenilich-Unendlichen. He felt, with some justice, 
that Scholastic thought had handled this subject more as a religious 
dogma than as a mathematical concept. 89 Moreover, the thought of 
Leibniz, whose calculus represented the most genial attempt to estab- 
lish a mathematics of the infinitely large and the infinitely small, 
lacked resolution. Sometimes he declared against the absolute infinite, 
and then again he remarked that nature, instead of abhorring the 
actual infinite, everywhere made use of it to mark better the perfec- 
tions of its author. 90 

The symbol <*> had been used by mathematicians since the time of 
Wallis to represent infinity, but no definition had been given, nor 
had the Scholastic distinction been observed. The symbol was used by 
Weierstrass, for example, in the sense both of a potentiality and of 

an actuality: he wrote /(a) = °° to mean that -— = 0, and also 

used the expression f( o° ) = b in the sense that the limit of f(x) for 
x indefinitely large was b 9l To avoid this confusion, Cantor chose a 
new symbol, «, to represent the actual infinite aggregate of positive 
integers. It is to be remarked, furthermore, that whereas o° referred 
in general to magnitude, co is to be interpreted in terms of aggregation. 
Wallis, Fontenelle, and others regarded oo variously as the largest 
positive integer or as the sum of all the positive integers; but the 
symbol &> refers to all the positive integers only in the sense that 
these form an aggregate of elements. This view of the infinite, as 
concerned with groups of elements, had previously been clearly ex- 
pressed by Bolzano, but he failed to recognize what Cantor called 
the power of an infinite set of elements. The rational numbers can 
be put into one-to-one correspondence with the positive integers, and 
for this reason these two classes are regarded as having the same 
power. One of the most striking results of Cantor's Mengenlehre, how- 
ever, is that there are transfinite numbers higher than «• The theory 
of arithmetic had shown that numbers other than the rationals were 
needed for the continuum, and Cantor now showed that the desired 
set of real numbers was such that these could not be put into one-to- 
one correspondence with the positive integers; i. e., the set was not 

89 Ibid., p. 191. . » Ibid., p. 179; see also Leibniz, Philosophised Schriften, I, 416. 
91 Cajori, A History of Mathematical Notations, II, 45. 



298 The Rigorous Formulation 

denumerable. It therefore represented a transfinite number of a higher 
power, which is often written now simply as C. Other numbers have 
been found above C, but the question as to whether there is one 
between a> and C is unanswered. However, the definition of the con- 
tinuous variable, and hence of the concepts of the calculus, requires 
only the infinite set C. 

Although Cantor's work does not clear up any objection raised on 
the ground of the conceptual difficulties inherent in the concept of 
infinity, it definitely refutes any argument raised upon the score of 
logical contradiction. Likewise, any criticism of the use of the infinite 
in defining irrational number or in the limit concept is answered by 
Cantor's work, which clarifies the situation, 92 As the terminus a quo 
of the investigations leading to the calculus is to be found in the 
Pythagorean discovery of the incommensurable and in the recognized 
need for satisfactory definitions of number and the infinite, so the 
terminus ad quern may be regarded as the establishment of these by 
the great triumvirate: Weierstrass, Dedekind, and Cantor. The 
fundamental notions of the calculus — the limit of a continuous vari- 
able: the derivative and the integral — have through the work of 
these men been given a logical rigor as impressive as that of Euclidean 
geometry, and a formal precision of which the Greeks had never 
dreamed. It has been shown that in analysis there is no need for any- 
thing but whole numbers, or finite or infinite systems of these. 93 
How startlingly apropos, with respect to the development of the 
calculus, is the Pythagorean dictum: All is number! 

82 It should, perhaps, be observed at this point that the theory of infinite aggregates 
has resulted also in a number of puzzling and as yet unresolved antimonies. See Poincare, 
Foundations of Science, pp. 477 ff.; Pierpont, "Mathematical Rigor," pp. 42-44. It has 
been suggested, in this connection, that these paradoxes may be related to the difficulties 
encountered in theoretical physics. See the review by Northrop, in Bulletin, American 
Mathematical Society, XLII (1936), 791-92, of Schrodinger, Science and the Human 
Temperament. 

93 PoincarS, The Foundations of Science, pp. 380, 441 ff. 



VIII. Conclusion 

THERE is a strong temptation on the part of professional mathe- 
maticians and scientists to seek always to ascribe great dis- 
coveries and inventions to single individuals. Such ascription serves 
a didactic end in centering attention upon certain fundamental aspects 
of the subjects, much as the history of events is conveniently divided 
into epochs for purposes of exposition. There is in such attributions 
and divisions, however, the serious danger that too great a significance 
will be attached to them. Rarely — perhaps never — is a single mathe- 
matician or scientist entitled to receive the full credit for an "innova- 
tion," nor does any one age deserve to be called the "renaissance" of 
an aspect of culture. Back of any discovery or invention there is 
invariably to be found an evolutionary development of ideas making 
its geniture possible. The history of the calculus furnishes a remarkably 
apt illustration of this fact. 1 

The method of fluxions of Newton was no more unanticipated than 
were his laws of motion and gravitation; and the differential calculus 
of Leibniz had been as fully adumbrated as had his law of continuity. 
These two men are to be thought of as the inventors of the calculus in 
the sense that they gave to the infinitesimal procedures of their 
predecessors the algorithmic unity and precision necessary for further 
development. Their work differed from the corresponding methods of 
their predecessors, Barrow and Fermat, more in attitude and general- 
ity than in substance and detail. The procedures of Barrow and 
Fermat were themselves but elaborations of the views of such men as 
Torricelli, Cavalieri, and Galileo, or Kepler, Valerio, and Stevin. The 
achievements of these early inventors of infinitesimal devices were in 
turn the direct results of the contributions of Oresme, Calculator, 
Archimedes, and Eudoxus. Finally, the work of the last-named men 
was inspired by the mathematical and philosophical problems sug- 
gested by Aristotle, Plato, Zeno, and Pythagoras. Without the filia- 
tion of ideas which was built up by these men and many others, the 
calculus of Newton and Leibniz would be unthinkable. 

1 Cf. Karpinski, "Is there Progress in Mathematical Discovery?" pp. 47-48. 



300 Conclusion 

If, on the one hand, mathematicians have been prone to forget the 
periods of suggestion and anticipation in the rise of the calculus, 
historians of the subject, on the other hand, have frequently failed to 
appreciate the significance of the later rigorous formulations. His- 
torical accounts of the subject all too often have terminated with the 
work of Newton and Leibniz, even though neither of these men was 
able to furnish the precision of thought which was to follow two 
centuries later. The neglect of this later period of investigation indi- 
cates an inadequate attention to, or appreciation of, the fundamental 
concepts of the subject as presented by Cauchy and Weierstrass. 
References, which might easily have been multiplied, have frequently 
been indicated above, in which the views expressed in the final elab- 
oration of the calculus have most unwarrantedly been imputed to 
earlier investigators in the field. Weierstrass' definition of real number 
has been identified with the theory of proportion of Eudoxus, and the 
Dedekind Cut with the speculations of Bryson. The continuum of 
Cantor has been viewed as expressed in the speculations of William 
of Occam or of Zeno. The limit concept of Weierstrass has been 
interpreted as identical with the prime and ultimate ratio of Newton, 
or even with the ancient Greek method of exhaustion. The derivative 
and the differential of Cauchy have been described as exactly corre- 
sponding to the related conceptions of Leibniz. The definite integral 
of Cauchy has been ascribed in all completeness to Fermat, or even 
to Cavalieri and Archimedes. 

Such citations make clear how general is the tendency unguardedly 
to read into the minds of earlier men one's own clear thoughts on the 
subject, forgetting that these are the culmination of centuries of 
speculation and investigation. The concepts of mathematics and sci- 
ence are eminently cumulative in their growth — the results of a con- 
tinuous effort to understand the relationships between elements and, 
in terms of these, to describe the confused impressions afforded by 
physical experience. The dynamics and astronomy of the sixteenth 
century, for example, were not entirely new developments, but rather 
grew out of medieval and ancient views on these subjects. In the same 
sense the use of infinitesimal conceptions during the early modern 
period did not proceed de novo, but began where the Scholastic 
philosophers and Greek mathematicians had left off. 



Conclusion 301 

Nevertheless, the fact of such progressive achievement is not to be 
interpreted as the unfolding of a well-conceived plan. Throughout the 
advance of science and mathematics, elements have constantly been 
discarded as well as added. For this reason no investigator is able to 
foresee the direction which the elaboration of his views will take. 
Only in retrospect can one trace the path along which such develop- 
ment has proceeded. Although in retracing this thread of thought 
one can readily recognize the notions from which the ultimate con- 
cepts have sprung, the former are not in general to be identified with 
the latter. Each is to be considered in the light of the mathematical 
and scientific milieu of the period in which it appeared. To interpret 
the geometric views of Archimedes and Barrow, for example, in terms 
of the analytical symbolism of the twentieth century is tantamount to 
invoking implicitly the precision and economy of thought which 
modern notations afford but which these earlier investigators were far 
from possessing. 

A deeper and more sympathetic understanding among professional 
workers in the fields of mathematics and history might easily remove 
much of the misdirected thought with respect to the nature and rise 
of mathematical concepts. A familiarity not only with the elements 
of the calculus, but also with the history of its development, will 
serve to bring out that the question is not so much who are the 
founders of the subject — Weierstrass and Cauchy, or Newton and 
Leibniz, or Barrow and Fermat, or Cavalieri and Kepler, or Archi- 
medes and Eudoxus — but rather in what sense each of these men 
may be regarded as responsible for the new analysis. 

It is possible not only to trace the path of development throughout 
the twenty-five-hundred-year interval during which the ideas of the 
calculus were being formulated, but also to indicate certain tendencies 
inimical to this growth. Perhaps the most manifest deterring force 
was the rigid insistence on the exclusion from mathematics of any idea 
not at the time allowing of strict logical interpretation. The very 
concepts which gave birth to the calculus — those of variation and 
continuity, of the infinite and the infinitesimal — were banned from 
Greek mathematics for this reason, the work of Euclid being a monu- 
ment to this exclusion. The work of Archimedes became most fruitful 
when he abandoned the Greek logical ideal and applied such forbidden 



302 Conclusion 

concepts, but this represents an almost isolated example of their use. 
Similarly, in the seventeenth century a number of mathematicians, 
including Pascal and Barrow, avoided the use of algebra and analytic 
geometry as not compatible with the demands of rigor inherited from 
the ancient Greeks. Had they made full use of these elements, they 
might well have been acclaimed the inventors of the calculus. Again 
in the eighteenth century English mathematicians disdained, largely 
because of the weak logical basis (as well as for reasons of national 
jealousy), to use the differential method of the Continental "com- 
putists," and consequently failed to make appreciable contributions 
to the rapid growth of analysis characterizing that century. 

It is clear that the indiscriminate use of methods and ideas which 
are palpably without logical foundation is not to be condoned. Such 
logical basis is, of course, ultimately to be sought in order to avoid 
hopeless confusion (as witness the eighteenth-century use of infinite 
series) ; but pending the final establishment of this, the banishment of 
suggestive views is a serious mistake. 

On the other hand, perhaps a more subtle, and therefore serious, 
hindrance to the development of the calculus was the failure, at vari- 
ous stages, to give to the concepts employed as concise and formal a 
definition as was possible at the time. The paradoxes of Zeno are 
excellent illustrations of the obscurity which results from a failure to 
specify clearly and unambiguously the conditions of the problem and 
to give formal definitions of the terms involved. Had the Greeks 
demanded of Zeno a precision of statement which the mathematicians 
exacted of themselves, they might not have banned the concepts lead- 
ing to the calculus nor disregarded almost entirely the science of 
dynamics. The nice distinctions of the Scholastic philosophers pointed 
the way to the clarification of such problems, but these men were not 
sufficiently familiar with the formalism of Greek geometry and Arabic 
algebra to be able to carry their ideas to completion. 

With the decline of Scholasticism, the tendency was away from 
precision of thought and toward the free use of imagination, as found 
in the literary Renaissance. The mathematical counterpart of this is 
seen strongly in the works of Nicholas of Cusa and Kepler, who 
employed in mathematics, without seeking adequately to define them, 
the conceptions of infinity, the infinitesimal, motion, and continuity. 



Conclusion 303 

In a sense this was fortunate, in that it favored the development of 
methods anticipating the calculus. On the other hand, however, the 
lack of a sound critical attitude was to leave undefined for hundreds 
of years the logical bases of the procedures thus employed, while the 
resulting confusion of thought engendered half a dozen alternative 
methods. Had Newton been more precise in the statement of his limit 
method, and had Leibniz been more explicit in professing that he was 
developing an instrument of invention and not a logical foundation, 
the period of indecision might not have ensued. As it was, it required 
the work of Cauchy, Weierstrass, and others to impart to the concepts 
of the continuous variable, the limit, the derivative, and the integral 
a precision of formulation which made them generally acceptable. 

In all probability, however, the chief obstacle in the way of the 
development of the concepts of the calculus was a misunderstanding 
as to the nature of mathematics. Ever since the empirical mathe- 
matics of the pre-Hellenic world was developed, the attitude has, 
upon occasion, been maintained that mathematics is a branch either 
of empirical science or of transcendental philosophy. In either case 
mathematics is not free to develop as it will, but is bound by certain 
restrictions: by conceptions derived either a posteriori from natural 
science, or assumed to be imposed a priori by an absolutistic phi- 
losophy. At the Egyptian and Babylonian level, mathematics was 
largely a body of information concerning the natural world. The early 
Ionians rearranged this knowledge into a deductive scheme, but the 
basis was still largely empirical science. The oriental mysticism of 
Pythagoras, however, reversed this state of affairs and gave to mathe- 
matics a supra-sensuous reality, of which the world of appearances 
was a counterpart. The premises were thus categorically established 
and all the mathematician could do was to develop the logical implica- 
tions of these. This view was elaborated by Plato into an idealistic 
philosophy which has consistently denied the purely logical and 
hypothetical nature of the propositions of mathematics. At the 
Greek stage of the ideas leading to the calculus, however, the views of 
Plato exerted a favorable influence, in that they counterbalanced the 
Peripatetic attitude. 

Aristotle considered mathematics an idealized abstraction from 
natural science, and as such the premises and definitions were not 



304 Conclusion 

arbitrary, but were determined by our interpretation of the world of 
sense perception. The concepts allowed to geometry were only such 
as were consistent with this picture. The infinitesimal and the actual 
infinite were excluded, not so much because of any demonstrated 
logical inconsistency, but because of a supposed incompatibility with 
the world of nature, from which the entities of mathematics were 
regarded as derived by a disassociation of irrelevant properties. Con- 
tinuity depended upon coincidence of extremities and upon oneness, 
as sense perception appeared to indicate. Likewise number was 
regarded, in conformity with the judgments of empiricism and com- 
mon sense, as a collection of units, with the result that irrational 
magnitudes were not considered as pertaining to the realm of number. 
Mathematics was the logic of relations, but the nature of these was 
completely determined by postulates which were in turn dictated by 
the evidence of physical experience. 

Of these two views, the Platonic and the Aristotelian, the former 
was for a time that under which the ideas of the calculus developed. 
Under this view, the conceptions of the infinite and the infinitesimal 
were not excluded, inasmuch as reason was not subject to the world 
of sensation. The entities of mathematics had an ontological reality, 
independent of common sense, and the postulates were discovered by 
reason alone. Although this made mathematics independent of natural 
science, it did not give it the postulational freedom it enjoys today. 

While the work of Archimedes displays elements of both the Platonic 
and the Aristotelian views, it was the latter which triumphed in the 
method of exhaustion and the classic geometry of Euclid. The con- 
cepts of infinity and continuity were consequently, during the medieval 
period, discussed from a dialectical rather than a mathematical point 
of view; but when they entered into the geometry of Nicholas of 
Cusa, of Kepler, of Galileo, it was largely under the Platonic view of 
rational transcendentalism, rather than of naturalistic description. 
Nevertheless, other mathematicians — Roberval, Torricelli, Barrow, 
and Newton — were led by the natural science of the day to interpret 
mathematics in terms of sense perception, as Aristotle had, and to 
introduce motion to avoid the difficulties of the infinite and the con- 
tinuous. The philosophers Hobbes and Berkeley felt the empirical 
tendency so strongly that they denied to mathematics the idealized 



Conclusion 305 

concept of a point without extension, because they felt that this had 
no counterpart in nature. 

The attitude of most of the mathematicians of the seventeenth 
century, however, was that of doubt. They employed infinitesimals 
and the infinite on the assumption that they existed, and treated the 
continuous as though made up of indivisibles, the results being 
justified pragmatically by their consistency with Euclidean geometry. 
In any case, the attitude was not that of an unprejudiced postulation 
and definition, followed by logical deduction. The investigations into 
the foundations of the calculus consequently took the form, during 
the eighteenth century, of a search for an explanation which should 
be intuitively plausible, rather than logically self -consistent. At this 
time, however, there was rapidly developing a very successful alge- 
braic formalism, vigorously fostered by Euler and Lagrange. This led, 
in the nineteenth century, to a view of mathematics which non- 
Euclidean geometry had strongly suggested — a postulational system 
independent alike of the world of sense experience and of any dictates 
resulting from introspection. The calculus became free to adopt its 
own premises and to frame its own definitions, subject only to the 
requirement of an inner consistency. The existence of a concept de- 
pended only upon a freedom from contradiction in the relations into 
which it entered. The bases of the calculus were then defined form- 
ally in terms only of number and infinite aggregates, with no corrob- 
oration through an appeal to the world of experience either possible 
or necessary. 

This formalizing tendency has not, however, been everywhere ac- 
cepted. Even mathematicians have not always been in sympathy with 
the movement. Hermite, whose favorite idea was to compare mathe- 
matics with the natural sciences, was horrified by Cantor's work, 
which transcended human experience. 2 Du Bois-Reymond similarly 
opposed the formal definitions which were made basic in the calculus 
and wished instead to define number in terms of geometric magnitude, 
much as Cauchy had done implicitly, 3 thus retaining intuition as 
a guide. More thoroughgoing intuitionists like Brouwer attempt 

2 See Poincare, "L'Avenir des mathematiques," p. 939. 

3 Pringsheim, "Nombres irrationnels et notion de limite," pp. 153 ff. See also 
Jourdain, "The Development of the Theory of Transfinite Numbers," 1913-14, pp. 
1, 9-10. 



306 Conclusion 

to visualize a fusion of the continuous and the discrete, somewhat 

as Plato had. 4 The mathematician Kronecker opposed the work of 

Dedekind and Cantor, not because of its formalism, but because he 

thought it unnatural. These investigators had "constructed" numbers 

which Kronecker felt could have no existence, and he proposed instead 

to base everything on equations involving integers alone, a view which 

has not been generally shared by other mathematicians. 5 

A number of scientists and philosophers have naturally been even 

more hesitant than these mathematicians in giving up experience and 

intuition in connection with the calculus. Thoroughgoing empiricists 

and idealistic philosophers in particular have sought, since the time 

of Newton and Leibniz to read into the calculus a significance beyond 

that of a formal postulational system. Newton had considered the 

calculus as a scientific description of the generation of magnitudes, 

and Leibniz had viewed it as a metaphysical explanation of such 

generation. The formalism of the nineteenth century took from the 

calculus any such preconceptions, leaving only the bare symbolic 

relationships between abstract mathematical entities. Nevertheless, 

traces of the old scientific and metaphysical tendencies remained. 

Lord Kelvin, who considered mathematics the etherealization of com- 

dx 
mon sense, once exclaimed, when he had asked what — represented 

at 

v A# 
and had received the answer lim — : "That's what Todhunter would 

At-*o Ai 

say. Does nobody know that it represents a velocity?" 6 His friend 
Helmholtz showed a similar tendency. In his famous essay, Ueber die 
Erhallung der Kraft, he regarded a surface as the sum of lines, 7 much 
as had Cavalieri in his Exercitationes just two centuries earlier. In 
another connection he asserted that incommensurable relations may 
occur in real objects, but that in numbers they can never be repre- 

4 See Helmholtz, Counting and Measuring, pp. xxii-xxiv; also Brouwer, "Intuitionism 
and Formalism," and Simon, "Historische Bemerkungen iiber das Continuum." 

5 See Couturat, De Vinfini matkematique, pp. 603 ff.; cf. also Pierpont, "Mathematical 
Rigor," pp. 38-40; Pringsheim, "Nombres irrationnels et notion de limite," pp. 
158-63; Jourdain, "The Development of the Theory of Transfinite Numbers," 1913-14, 
pp. 2-8. 

6 Hart, Makers of Science, pp. 278-79; Felix Klein, Entwicklung der Mathematik im 19. 
Jahrhundert, I, 238. 

7 Ueber die Erhaltung der Kraft, p. 14. 



Conclusion 307 

sented with exactness. 8 Mach also felt strongly the empirical origin of 
mathematics and held with Aristotle that geometric concepts are the 
product of idealization of physical experiences of space. 9 In conformity 
with this view, he felt that some form of geometrical meaning had 
necessarily to be given to the number i. 10 In this respect he is in agree- 
ment with a number of present-day scientists, who feel that V — 1 
simply "forms a part of various ingenious devices for handling other- 
wise intractable situations." 11 

The attitudes of Helmholtz and Mach are representative of the 
influence in science of the positive philosophy of the nineteenth cen- 
tury. Positivistic and materialistic thought were slow to accept the 
changed mathematical view and insisted that the calculus be inter- 
preted in terms of velocities and actual intervals, corresponding to 
the data of experience and of ordinary algebra. Comte recognized that 
mathematics is not "the science of magnitudes," 12 but he did not rise 
to the formal view of Cauchy. In accordance with the empirical and 
pragmatic attitude of Carnot, he regarded the methods of Newton, 
Leibniz, and Lagrange as fundamentally identical. However, because 
the differential calculus gave no clear conception of the infinitely 
small and because the method of limits apparently separated the fields 
of ordinary and transcendental analysis, he felt that the method of 
Lagrange was to be favored. 13 More strongly expressed are the views of 
Duhring, who, in 1872, in his classic Kritische Geschichte der allgemeinen 
Principien der Mechanik, indulged in a polemic against Gauss, Cauchy, 
and others who would deny the absolute truth of geometry, and who 
would introduce into mathematics such figments of the imagination 
as imaginary numbers, non-Euclidean geometry, and limits! 14 Marxian 
materialists will not grant mathematics the independence of experi- 
ence necessary for its proper development. 15 Such denial makes impos- 
sible the concept of the derivative and the scientific description of 
motion in terms thereof. The mathematical infinity is, in accord with 

8 Helmholtz, Counting and Measuring, p. 26. 

9 Mach, Space and Geometry, p. 94; cf. also p. 67. See also Strong, Procedures and 
Metaphysics, p. 232. 

xo Space and Geometry, p. 104, n. u Heyl, "The Skeptical Physicist," p. 228. 

12 Comte, The Philosophy of Mathematics, p. 18. a Ibid., pp. 110-17. 

14 Duhring, Kritische Geschichte der allgemeinen Principien der Mechanik, pp. 475 ff ., 
529 ff. 

15 Engels, Herr Eugen Diihring's Revolution in Science, p. 47. 



308 Conclusion 

this view, a contradiction of the "tautology" that the whole is greater 
than any of its parts. 16 

If a number of philosophers were led by excessive realism to reject 
much of the mathematics of the nineteenth century, idealistic phil- 
osophers, following Kant, were likewise unwilling to accept the bare 
formalism of Cauchy and Weierstrass in the realm of the calculus. 
The differential had been defined by Cauchy, not as a fixed quantity, 
but as a variable, and Weierstrass had shown that the continuous 
variable depends only upon the static notion of sets of elements. 
Idealists attempted, nevertheless, to interpret the differential as hav- 
ing an intensive quality resembling the potentiality of Aristotle, the 
impetus of the Scholastics, the conatus of Hobbes, or the inertia of 
modern science. They wished to view the continuum, not in terms 
of the discreteness of Cantor and Dedekind, but as an unanalyzable 
concept in the form of a metaphysical reality which is intuitively 
perceived. The differential calculus was regarded as possessing a 
"positive" meaning as the generator of the continuum, as opposed to 
the "negation" of the limit concept. 17 As Hegel expressed it, the 
derivative represented the "becoming" of magnitudes, 18 as opposed to 
the integral, or the "has become." 

Materialistic and idealistic philosophies have both failed to appre- 
ciate the nature of mathematics, as accepted at the present time. 
Mathematics is neither a description of nature nor an explanation of 
its operation; it is not concerned with physical motion or with the 
metaphysical generation of quantities. It is merely the symbolic logic 
of possible relations, 19 and as such is concerned with neither approxi- 
mate nor absolute truth, but only with hypothetical truth. That is, 
mathematics determines what conclusions will follow logically from 
given premises. The conjunction of mathematics and philosophy, or 
of mathematics and science, is frequently of great service in suggesting 
new problems and points of view. 

u Ibid., pp. 48-49, 62; cf. also Bois, "Le Finitisme de Duhring," p. 95. 

17 See Kant, Sammtliche Werke, XI (Part I), 270-71; cf. also II, 140-49 and passim. 
See also Cohen, Die Princip der Infinitesimalmethode und seine Geschichte, passim; Simon, 
"Zur Geschichte und Philosophic der Differentialrechnung," p. 128; Vivanti, "Note sur 
l'histoire de I'infiniment petit," pp. 1 ff.; Freyer, SPudien zur Metaphysik der Differential- 
rechnung, pp. 23 ff.; Lasswitz, Geschichte der Atomistik, I, 201. 

M Klein, Elementary Mathematics from an Advanced Standpoint, p. 217. 

18 Cohen, M. R. } Reason and Nature, pp. 171-205. 



Conclusion 309 

Nevertheless, in the final rigorous formulation and elaboration of 
such concepts as have been introduced, mathematics must necessarily 
be unprejudiced by any irrelevant elements in the experiences from 
which they have arisen. 20 Any attempt to restrict the freedom of choice 
of its postulates and definitions is predicated on the assumption that 
a given preconceived notion of the nature of the relationships involved 
is necessarily valid. The calculus is without doubt the greatest aid 
we have to the discovery and appreciation of physical truth; but the 
basis for this success is in all probability to be found in the fact that 
the concepts involved were gradually emancipated from the qualitative 
preconceptions which result from our experiences of variability and 
multiplicity. Greek philosophy had attempted to separate and con- 
trast the qualitative and the quantitative, but the later medieval and 
early modern period associated them through geometric representation. 
Even a quantitative explanation is subject to sensory notions of size, 
length, duration, and so forth, so that greater independence was 
achieved in the nineteenth century by basing the calculus upon 
ordinal considerations only. The history of the concepts of the calculus 
shows that the explanation of the qualitative is to be made through 
the quantitative, and the latter is ^n turn to be explained through the 
ordinal, perhaps the most fundamental notion in mathematics. As the 
sensations of motion and discreteness led to the abstract notions of 
the calculus, so may sensory experience continue thus to suggest prob- 
lems for the mathematician, and so may he in turn be free to reduce 
these to the basic formal logical/ relationships involved. Thus only may 
be fully appreciated the twofold aspect of mathematics: as the lan- 
guage of a descriptive interpretation of the relationships discovered 
in natural phenomena, and as a syllogistic elaboration of arbitrary 
premises. 

20 Cf . Poincar6, Foundations of Scienci, pp. 28-29, 46, 65, 428. 



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Index 



Abel, N. H., 280 

Academie des Sciences, 241, 242, 263 

Acceleration, Greek astronomy lacked con- 
cept of, 72; uniform, 82 ff., 113ff., 130, 
165 

Achilles, 24-25, 116, 138, 140, 295; argu- 
ment in the, 24«; refuted by Aristotle, 44 

Acta eruditorum, 207, 208, 214, 221, 238 

A est unum calidum, 87 

Agardh, C. A., Essai sur la melaphysique dn 
calcul integral. 264 

Albert of Saxony, 66, 68 f., 74 

Alembert, Jean leRond d', 237; concept of 
differentials, 246, 275; interpretation of 
Newton's prime and ultimate ratio, 247; 
quoted, 248; limit concept, 249, 250, 253, 
257, 265, 271, 272; definition of the 
tangent, 251; concept of infinitesimal, 
274; concept of infinity, 275 

Algebra, Arabic development, 2, 56, 60, 
63 ff., 97, 120, 154; contribution of 
Hindus, 63; symbols for quantities, 98; 
use of, avoided in seventeenth century, 
302 

Algebraic formalism fostered by Euler and 
Lagrange, 305 

Alhazen (Ibn al-Haitham), 63 

Ampere, A. M., theory of functions, 282 

Anaxagoras, 40 

Anaximander, 28, 40 

Angle of contact, 22, 173, 174, 212 

Antiphon the Sophist, 32 

Apeiron, 28 

Apollonius, influence in development of 
analytic geometry, 187 

Arabs, algebra adopted from, 2; extended 
work of Archimedes, 56, 120; algebraic 
development, 60, 63 ff., 97, 154 

Arbogast, L. F. A., Du calcul des derivations, 
263 

Archimedes, 25, 29, 38, 39, 48, 49-60, 
64, 65, 70, 101, 105, 112, 299, 
Method, 21, 48-51, 59, 99, 125, 139, 
159; axiom of, 33, 173; use of infinitely 
large and infinitely small, 48 ff., 90;. 
quadrature of the parabola, 49-53, 102, 
124; volume of conoid, 53-55; spiral, 55- 
58, 133; determination of tangents, 56 ff.; 



work on centers of gravity, 60; use of 
series, 76 f., 120; mensurational work, 89, 
94, 107; translation and publication of 
works, 94, 96, 98; modifications of 
method of, Chap. IV; use of doctrine of 
instantaneous velocity, 56 ff., 130; in- 
equalities, 150 

Area, concept of, 9, 31, 34, 58f., 62, 105, 
123; application of, 18, 19, 32, 36, 96 

Aristotle, 1, 20, 22, 26, 30, 46, 57, 60, 65, 66, 
67, 70, 72, 79, 112, 299, 303; opposed to 
Plato's concept of number, 27; On the 
Pythagoreans, 37; dependence upon 
sensory perception, 37 ff.; considered 
mathematics an idealized abstraction 
from natural science, 37-38, 303, 307; 
quoted, 38, 41, 42, 43, 46; concept of the 
infinite, 38ff., 69, 70, 102, 117, 151, 275, 
296; concept of number, 40-41, 151, 154- 
55; Physica, 41, 43, 65, 78; theory of 
continuity, 41-42, 66-67, 277, 291; con- 
cept of motion, 42-45, 71 ff., 176, 178, 
233; denial of instantaneous velocity, 43, 
73, 82; logic of, 45, 46, 47, 61; sixteenth- 
century opposition to Aristotelianism, 96; 
views triumphed in method of exhaustion 
and geometry of Euclid, 45 ff., 304 

Arithmetic triangle, see Pascal's triangle 

Arrow, argument in, 24», 295; refuted by 
Aristotle, 44 

Assemblages, see Infinite aggregates 

Athens, 26 

Atomic doctrine, 21, 23, 38, 42, 66, 67, 84, 
92, 180, 181 

Atomism, mathematical, 22, 28, 50, 82, 
115, 125, 134, 176, 188 

Average velocity, 6, 8, 75, 83, 113 ff. 

Babylonians, mathematical knowledge, 1, 
14 ff., 62, 303; astronomical knowledge, 
15 

Bacon, Roger, 64; Opus majus, 66; concept 
of motion, 72 

Barrow, Isaac, 58, 133, 209, 299, 302, 304; 
anticipated invention of the calculus, 164; 
opposed arithmetization of mathematics, 
175; advocated classic concept of number 



338 



Index 



Barrow, Isaac {Continued) 
and geometry, 179; distrust of algebraic 
methods, 180; concepts involved in his 
geometry, 181; quoted, 182; Geometrical 
Lectures, 182, 185, 189, 202; method of 
tangents, 182-86, 192-93, 203, 210, 214; 
influence in development of the calculus, 
187; concept of time, 194, 233, 295; dif- 
ferential triangle, 183, 203 

Bayle, Pierre, 217, 224 

Beaugrand, Jean de, 120 

Bede, 66 

Berkeley, George, 156, 257 f.; criticism of 
Newton, 224 ff.; The Analyst, 224-29, 
232, 233, 235; Essay towards a New 
Theory of Vision, 224; influence of 
empirical tendency, 227, 304; A Defence 
of Freethinking in Mathematics, 228; 
Berkeley - Jurin - Robins controversy, 
229 ff., 246, 263 

Berlin Academy, 254 

Bernoulli, James, 153, 207, 220, 238; con- 
cept of infinitesimal, 239 

Bernoulli, John, 207, 211, 212, 220, 238, 
246; concept of infinitesimal, 239, 240; 
treatment of infinite and infinitesimal, 
242; concept of the integral, 278 

Bhaskara, 62, 192 

Binomial theorem, 119, 148, 190, 191, 194, 
207, 225 

Blasius of Parma (Biagio Pelicani), 84, 88, 
89 

Boethius, Geometry, 64 

Bolzano, Bernhard, 268, 284; concept of 
infinity, 116, 270, 274f., 288, 296, 297; 
Rein analytischer Beweis des Lehrsatzes 
. . ., 266; limit concept, 269; need for 
definitions of convergence of series, 270, 
280; theory of functions, 276, 282; 
definition of continuity, 277, 279 

Bombelli, Raphael, 97 

Boyle, Robert, 2 

Bradwardine, Thomas, 66fL, 97, 116; Geo- 
metria speculativa, and Tractatus de con- 
tinuo, 66; quoted, 67; Liber de propor- 
tionibus, 69, 74 

Brahmagupta, 62 

Brasseur, J. B. "Exposition nouvelle des 
principes de calcul differentiel," 264 

Brouwer, L. E. J., 28, 67, 305; see also 
Intuitionism 

Bruno, Giordano, 107 

Bryson of Heraclea, 32, 33, 300 



Buffon, Comte de, 246, 247 

Buquoy, Georg von, Eine neue Methodefiir 

den Infinitesimalkalkul, 264 
Buridan, Jean, 72, 113; doctrine of impetus 

(or inertia), 177 
Burley, Walter, 66, 74 



Calculator, see Suiseth, Richard 

Cambridge Platonists, 180 

Cantor, Georg, 33, 271, 284; sought defini- 
tion of irrational number, 288; "tiber die 
Ausdehnung eines Satzes aus der Theorie 
der trigonometrischen Reihen," 289; 
Mengenlehre, 296, 297; concept of the 
infinite, 296, 297, 298; continuum of, 
296 ff., 300 

Cantor-Dedekind axion, 291-92 

Capella, Martianus, 66 

Cardan, Jerome, 97, 118; De subtilitate, 88 

Carneades, 38 

Carnot, L. N. M., 226; doctrine of perfect 
and imperfect equations, 158; Reflexions 
sur la metaphysique du calcul infinitSsi- 
mal, 257; concept of infinitesimal, 258, 
283 

Cauchy, A. L., Ill, 123, 268; differential 
defined in terms of the derivative, 210, 
275, 276; theorem of, 270, 281, 282, 289; 
Cours d' 'analyse, 271; excerpt, 272, 281; 
Resume des legons sur le calcul infinitesi- 
mal, 271; Legons sur le calcul diffirentiel, 
271; definition of limit, 272, 287; ideas 
of variable, function, limit, and orders of 
infinity, 272 ff., 281; concept of infinitesi- 
mal, 273; definition of continuity, 277, 
279 

Cavalieri, Bonaventura, 28, 48, 56, 109, 151, 
160, 204, 299; theorem of, 22, 118; 
Geometria indivisibilibus, 111, 112, 117, 
119, 121, 122, 123; forces molding 
thought of, 112, 117 ff.; Exercitationes 
geometricae sex, 117, 135, 306; role in 
development of infinitesimal methods, 
121, 123; method of indivisibles, 125, 141, 
142, 150, 151, 170, 181, 195, 227 

Characteristic triangle, see Differential 
triangle 

Circle, conception of as a limit, 32, 231, 
272 

Collins, John, 185 

Commandino, Federigo, 99, 101, 112, 159; 
Liber de centro gravitaiis solidorum, 94 



Index 



339 



Commercium episiolicutn, 221, 222 

Compensation of errors, 226, 245, 251, 
257 

Comte, Auguste, 307 

Conatus, Hobbes's concept of, 43, 178, 179, 
212, 308 

Condorcet, N. C. de, 263 

Continuity, concept of, 3, 4, 25, 26, 29, 35, 
37, 38, 41, 44, 47, 51, 62, 63, 65-68, 116, 
151, 152, 155, 179, 181, 184, 226, 229, 
246, 267-70, 279, 282, 288, 291, 292, 294, 
295; Aristotle's conception, 41-45, 304; 
medieval conception, 65 ff., 304; law of, 
93, 110, 116, 172, 178, 217-22 passim, 
244, 245, 256, 258, 268, 277, 281, 283, 294, 
299; Leibniz's doctrine of, 217, 268, 277, 
291; definition of, 268, 277, 279, 286, 
294, 298 

Continuum, 3, 4, 13, 16, 17, 20, 42, 43, 45, 
60, 66, 67, 115, 116, 179, 181, 217, 218, 
256, 270, 291; generated by flowing of 
apeiron, 28; mathematical concept of, 
35, 47, 92, 227; attempted link with 
indivisibles, 91; Barrow's treatment of, 
180; views of Cantor imputed to earlier 
investigators, 300; as an unanalyzable 
concept, 308 

Convergence of series, 174, 207, 253, 264, 
270, 280 ff. 

Cournot, A. A., Traite elementaire de la 
theorie des fonctions et du calcul infinitesi- 
mal, 283 

Crelle's Journal, see Journal fiir die Reine 
und Angewandte Malhematik 

Cusa, Nicholas of, see Nicholas of Cusa 

Cycloid, 151, 167 



Dedekind, Richard, basis of definition of 
infinite assemblages, 271; Sietigkeit und 
die Irrationalzahlen, 289; theory of ir- 
rational numbers, 290 ff.; concept of the 
infinite, 296 

Dedekind Cut, 33, 42, 292, 294, 300 

Definite integral, see Integral, definite 

De lineis insecabilibus, 39 

Democritus, 21, 22, 26, 28, 30, 37, 39, 48, 
66, 71, 91, 108; see also Atomic doctrine; 
Atomism, mathematical 

Derivative, concept of, 6, 13, 22, 28, 40, 
43, 47, 59, 61, 63, 79, 80, 82, 85, 111, 157, 
179, 184, 185, 186, 191, 196, 251, 255, 
259, 262, 277, 282, 295, 298, 300, 308; 



defined, 3, 4, 5, 7, 8, 11, 12, 269, 275, 
293, 294; differential defined in terms of, 
210, 275, 276; origin of word, 252-54; 
continuous functions and, 269, 270, 
279 

Derived functions, Lagrange's method of, 
251-55, 259-65, 267, 270, 272, 275, 278; 
see also Derivative 

Descartes, Rene, 2, 29, 58, 88, 133, 165, 
238; Geometrie, 154, 166, 208; concept of 
variability, 154, 155; method for normal 
to a curve, 158; infinitesimal concept, 
165, 168, 193; method of tangents, 166- 
68; concept of instantaneous center of 
rotation, 167; notion of instantaneous 
velocity, 168, 177; Discours de la 
methodc, 168; analytic geometry, 154, 
165, 179, 187; number concept, 242; 
method of undetermined coefficients, 166, 
259 

Dichotomy, 116, 295; argument in, 24w; 
argument refuted by Aristotle, 44 

Difference, see Differential 

Differences, method of, see Differentials, 
method of 

Differential coefficient, 276; origin of term, 
253, 265; see also Derivative 

Differential quotient, 263, 275; integral the 
inverse of, 283 ; see also Derivative 

Differentials, 111, 123, 150, 173, 179, 187- 
223, 224, 278; defined, 12, 210, 259, 275, 
276, 308; method of. 59, 122, 145, 178, 
185, 187-223, 224, 226, 234, 236ff., 243, 
244 ff., 251, 253, 266, 267, 277, 294, 299, 
300, 302; origin of word, 205-6 

Differential triangle, 152, 163, 182, 203, 218, 
242 

Diodes, curve defined by motion, 71 

Diogenes Laertius, 22 

Diophantine equations, 72 

Diophantus, 63; Arithmetic, 60 

Distance, represented by area under 
velocity-time curve, 82-84, 113-15, 126, 
130-31, 134, 180 

Divergent series, 207, 241, 245, 246, 265, 
280 

Du Bois-Reymond, Paul, 305 

Diihring, Eugen; Kritische Geschichte der 
allgemeinen Principien der Mechanik, 307 

Duns Scotus, John, 66, 74 

Dynamics, 45, 178; Galilean, 72, 82, 83, 
113-15, 130, 177-78; medieval, 72 ff., 
82 ff. 



340 



Index 



Egyptian mathematics, 1, 14ff., 21, 303 

Eleatics, 23 

Encyclopedia Britannica, excerpt on fluxion, 
232 

Epicurus, 22, 91 

Eratosthenes, 48 

Euclid, 1, 2, 22, 27, 30, 48, 61, 64, 90, 173; 
Elements, 20, 45 ft"., 64; debt to Eudoxus, 
30; quoted, 31, 33, 240; method of 
exhaustion, 34, 41; logic, 46 ff.; view of 
number, 47, 221; banned concepts which 
gave birth to the calculus, 47-48, 301; 
concept of tangent, 57 

Eudoxus, 30 ff., 37, 45, 48, 299; method of 
exhaustion, 30, 31, 34, 41, 46, 96; defini- 
tion of proportion, 31, 47, 300 

Euler, Leonhard, 167, 224, 226, 234, 250; 
concept of differentials, 150, 253; func- 
tion concept, 243, 272, 275; view of the 
infinite, 245, 249; divergent series, 246, 
280; concept of the integral, 278 

Evanescent quantities, 216, 228 

Exhaustion, method of, 4, 9, 26, 30, 31, 
32 ff., 36, 41, 46, 48, 51 ff., 62, 95, 99, 100, 
104, 106, 109, 123, 124, 128, 133, 136, 150, 
171, 197, 215, 255, 258, 259, 271, 300, 
304; origin of word, 34, 136, 140; concept 
of infinity substituted for method of, 
142 



Faille, Jean-Charles della, 138 

Fermat, Pierre de, 48, 56, 120, 121, 125, 
127, 140, 142, 154, 164, 168, 169, 175, 187, 
208, 209 f., 299, 300; method of maxima 
and minima, 111, 155-59, 163, 164, 185- 
86, 202; method of tangents, 157, 162 ff., 
167, 175, 183 ff., 192, 202; quadratures, 
159-64, 202; use of infinite progressions, 
160-61, 190; concept of definite integral, 
161-62, 173; rectification of curves, 162; 
differential triangle, 163, 203; Opera 
varia, 164 

Ferrari, Lodovico, 97 

Fluent, 79, 187-223 

Fluxions, 79, 173, 187-223, 224, 226, 232, 
250, 260, 278; method of, 58, 59, 122, 129, 
178, 187-223, 224, 225, 226, 229, 236, 237, 
238, 243, 246, 247, 251, 253, 263, 265, 266, 
267, 294, 299; views of Robins, Maclau- 
rin, Taylor and Simpson, 232 ff. 

Fonctions derivees, see Lagrange, J. L., 
method of derived functions 



Fontenelle, Bernard de, 214, 241, 249; 

treatment of infinite and infinitesimal, 

242; EUmens de la glometrie de Vinfini, 

242 
Formalism, see Mathematical formalism 
Forms, latitude of, 73 ff., 82, 85, 86, 87, 88, 

126; graphical representation, 81-84, 111 
Fourier, J. B., function concept, 276 
Function concept, 55, 56, 58, 60, 94, 98, 

156, 184, 186, 196, 220, 221, 236, 237, 

240, 243, 259, 272, 273, 276 
Fundamental theorem of the calculus, 10, 

11, 164, 181, 184, 187, 191, 194, 196, 203, 

204 ff., 279 



Galileo, 2, 29, 70, 121, 177, 180, 299; con- 
cept of the infinite, 70, 115 ff., 193, 270; 
dynamics, 72, 82, 83, 89, 113 ff., 133; 
recognition of law of uniform accelera- 
tion, 82, 85, 113ff., 130; concept of area, 
84, 134; Two New Sciences, 85, 112, 113, 
115; forces molding thought of, 112 ff.; 
idea of impetus, 113, 130, 177; concept of 
the infinitely small, 115 ff., 195; doctrine 
of continuity, 116, 291 

Gassendi, Pierre, 224 

Gauss, K. F., 98, 268, 280 

Geometry, Greek, see Greeks 

Gerdil, Hyacinth Sigismund, 250 

Girard, Albert, 104 

Goethals, Henry, 66, 74 

Grandi, Guido, 218; interest in dif- 
ferential calculus, 238; concept of in- 
finitesimal, 241 

Graphical representation of variables, 80 ff., 
Ill, 113ff., 190, 288, 309 

Gravity, centers of, 60, 93, 94, 99 ff., 104, 
139, 145, 158, 163, 166 

Greeks, search for universals, 8, 14; devel- 
opment of mathematics and science, 
16 ff.; geometry, 17, 26, 46, 65; concept of 
number, 18, 29, 41, 43; position of Zeno's 
paradoxes in thought of, 25; concept of 
length, area, and volume, 31 ff.; concept 
of proportion, 31 ff.; method of exhaus- 
tion, 33S., 35, 100, 136, 197, 271, 300; 
concept of motion, 42 ff., 71, 132; 
skeptics, 46, 214; abandonment of at- 
tempt to associate numbers with all 
geometric magnitudes, 20, 62; mathe- 
matics in Middle Ages, 64; concept of 
instantaneous velocity, 43, 57, 82 



Index 



341 



Gregory, James, 192, 280; Vera circuit et 
hyperbolae quadratura, 174; arithmetical 
and analytical work, 175; quadratures, 
175; use of Fermat's tangent method, 
175, 183 

Gregory of Rimini, 66, 69 

Gregory of St. Vincent, 162, 202; Opus 
geometricum, 135, 160, 169; "ductus 
plani in planum," 135, 145; view of 
nature of infinitesimals, 135 ff.; use of 
word "exhaust," 136; limit concept, 137, 
152, 169, 282; geometric progressions, 
137-38, 174, 190; influence, disciples, 138; 
method of indivisibles, 141 

Grosseteste, Robert, 66 

Guldin, Paul, 121, 138 

Guldin's Rule, see Pappus, theorem 



Hadamard, J., 276 

Halley, Edmund, 222, 224 

Hankel, Hermann, 271 

Harmonic triangle, 204 

Harriot, Thomas, 169 

Hegel, G. W. F., 308 

Heine, Eduard, 290; limit concept, 287; 
"Die Elemente der Funktionenlehre," 
288 

Helmholtz, Hermann von, 294, 307; Die 
Erhaltung der Kraft, 306 

Hentisbery, William of, 84, 88, 96, 113, 
177 

Heraclitus, 25, 71 

Hermite, Charles, 305 

Heron of Alexandria, 59, 115 

Heterogenea, 140, 152 

Heuraet, Heinrich van, rectification of 
curves, 162 

Hindus, Pythagoras' debt to, 19; mathe- 
matics, 60, 61 ff., 72, 97 

Hipparchus, 59, 63 

Hippias of Elis, 56, 71 

Hippocrates of Chios, 33 

Hobbes, Thomas, 212, 227; concept of the 
conatus, 43, 178, 179, 195, 308; opposed 
arithmetization of mathematics, 175; 
concept of number and of the geometrical 
elements, 176; idea of motion, 178 

Homogenea, 140 

Horn angle, see Angle of contact 

Hudde, Johann, rules for tangents and for 
maxima and minima, 185 f. 

Humanism, 89 



Huygens, Christiaan, 153, 208, 213; use of 
Fermat's tangent method, 183, 186; rules 
for tangents and for maxima and minima, 
185 f. 



Ibn al-Haitham, see Alhazen 

Impetus (inertia), concept of, 43, 72, 130, 
177 

Incommensurable, see Irrational 

Indefinite integral, see Integral, indefinite 

Indivisibles, 11, 13, 22, 27, 38, 63, 66 ff., 
84,91, 109, 116, 118ff., 123, 124, 151, 173, 
197, 227; method of, 50, 56, 92, 117-26, 
133, 134, 139, 140, 141, 142, 143 ff., 149 ff., 
181 

Inertia, see Impetus 

Infinite aggregates, 13, 25, 68, 70, 115, 249, 
257, 268, 270, 275, 277, 284, 286, 290, 291, 
293-98; definition, 296 

Infinitely small, see Infinitesimal 

Infinite sequence, 4, 7, 8, 9, 11, 24, 36, 37, 
50, 53, 102; see also Infinite series 

Infinite series, 24, 44, 52, 55, 56, 76, 86, 116, 
126, 137, 140, 142, 152, 154, 173, 190, 
192, 199, 216, 221, 226, 231, 245, 254, 
256, 264, 265, 267, 268, 270, 276, 280, 
281, 282, 284, 286, 289, 290, 302; con- 
vergence of, 174, 207, 253, 264, 270, 
280 ff. 

Infinitesimal, 7, 11, 12, 22, 29, 50, 51, 55, 
59, 62, 63, 65, 68, 83, 84, 90, 92, 96, 108, 
109, 114, 115, 117, 135, 142, 145, 152, 
162, 163, 166, 169, 176, 180, 181, 183 ff., 
187, 191, 193, 195, 196, 197, 199, 200, 
201, 202ff., 209ff., 212, 215, 218, 220, 
224, 225, 229 ff., 258, 260, 263, 265, 286, 
299, 301; concept of, 4, 21, 28, 39, 47, 
154 ff., 213 ff., 239 ff., 262, 300 {see also 
under Cavalieri; Descartes; Pascal); not 
found before Greek period, 15; Zeno's 
dictum against, 23; Democritean view, 
30; Aristotelian view, 38 ff.; as an in- 
tensive quantity, 43, l78f., 262; Archi- 
medes' view, 49 ff.; medieval views, 66 ff.; 
potential existence, 67; view of Nicholas 
of Cusa, 90 ff.; Kepler's view, 108 ff.; 
Galileo's view, 112 ff.; view of Gregory of 
St. Vincent, 135 ff.; Tacquet's view, 
139 ff.; Roberval's view, 141 ff.; Pascal's 
view, 149 ff.; Fermat's view, 154 ff.; 
Wallis' view of, 170ff.; philosophers 
reluctant to abandon, 179; Barrow's 



342 



Index 



Infinitesimal (Continued) 
view, 180 ff.; Newton's view, 193 ff.; 
Leibniz's view, 209 ff.; confused with 
fluxions, 223; Jurin's view, 229; Robins' 
disavowal of, 230; views of John and 
James Bernoulli, 238 ff.; views of Wolff 
and Grandi, 240-41; view of Fontenelle, 
241-42; Euler's view, 244-46; definition, 
248, 273; D'Alembert's view, 248f.; views 
of Kastner and Lagrange, 250 ff.; Car- 
not's view, 258 f.; views of Bolzano and 
Cauchy, 270 ff. 

Infinitesimals, fixed, see Indivisibles 

Infinity, 2, 6, 22, 38, 41, 44, 51, 52, 56, 63, 
65, 66, 70, 76, 77, 90, 91, 92, 108, 121, 
143, 151, 152, 172, 213, 218, 239ff., 267; 
concept of, 25, 39, 47, 53, 115ff., 226, 
268, 296, 302, 304; Aristotle's view, 38- 
41, 68; actual and potential, 40, 41,' 68, 
69, 77, 102, 117, 151; banned from Greek 
mathematics, 46, 47, 96, 142, 301; 
Scholastic view, 68 ff.; Galileo's view, 70, 
115, 193, 270; mathematical concept, 116, 
170-71, 227, 270-71, 274-75, 284, 296ff.; 
Bolzano's view, 116, 270-71, 296; sym- 
bols used to represent, 170, 297; orders of, 
245, 274; Cauchy's view, 274-75, 284, 
296; view of Dedekind and Cantor, 296 ff. 

Inscribed and circumscribed figures, propo- 
sition of Luca Valerio, 124, 128, 137 

Instantaneous center of rotation, 167 

Instantaneous direction, 57f., 132, 134, 146, 
189, 212 

Instantaneous rate of change, 78, 81, 82, 83 

Instantaneous velocity, 6, 7, 8, 43, 44, 47, 
59, 73, 82, 84, 115, 116, 130 ff., 168, 178, 
179, 180, 191, 193 ff., 200, 221, 227, 232, 
233, 295; quantitative treatment, 73 ff., 
82 ff., 113 ff., 130 ff., 177; medieval view, 
73 ff.; Barrow's treatment, 180; Newton's 
view, 194 ff.; views of Robins, Mac- 
laurin, Taylor, and Simpson, 234 

Instants, 92 

Integral, 22, 40, 80, 109, 123, 161, 187-223 
293, 294, 298; defined, 3, 5, 11, 12, 206, 
239, 278, 279, 294; concepts of, 6, 13, 47, 
56, 61, 63, 184, 185, 295, 308; origin of 
word, 67, 205, 206 

Integral, definite, 8, 50, 56, 120, 121, 123, 
127, 128, 143, 144, 146, 149, 152, 159, 
161, 191, 206, 278; defined, 9, 10, 55, 279; 
concept of, 142, 173 

Integral, indefinite, 191, 206, 278; defined, 1 1 



Integral of Lebesque, 280 

Integration, by parts, 152, 163 

Intensity, average, 74-78 

Intensive magnitude, 28, 43, 177, 178, 179, 

262, 308 
Intuition, 5; Euclid's Elements based on, 47; 

of uniform rate of change, 78 
Intuitionism, mathematical, 3, 28, 67, 209, 

305 
Ionians, 18, 21; empirical science the basis 

of mathematics of, 303 
Irrational and incommensurable, 18 ff., 20, 

31 ff., 35, 37, 61 ff., 66, 96, 97, 107, 121, 

173, 174, 176, 179, 190, 256, 281, 282, 

284, 285, 288-94, 296, 300, 306 
Isidore of Seville, 66 

James of Forli, 74 

John XXI, pope (Petrus Hispanus), Sutn- 
mulae logicales, 68 

Jordanus Nemorarius, 64, 96, 98, 173 

Journal des savants, 238 

Journal fiir die Reine und Angewandte 
Mathematik, 288 

Jurin, James, Geometry No Friend to In- 
fidelity, 228; The Minute Mathematician, 
228; attitude toward infinitesimal, 229; 
controversy with Berkeley and Robins, 
229 ff., 246, 263 

Kant, Immanuel, 2, 3, 261, 264, 308 

Kastner, A. G., 254; Anfangsgriinde der 
Analysis des Unendlichen, 250 

Kelvin, William Thomson, baron, quoted, 
306 

Kepler, Johann, 2, 48, 91, 94, 156, 299; 
influenced by Nicholas of Cusa, 93; 
speculative tendency, 106; Mysterium 
cosmographicum, 107; work on curvilinear 
mensuration, 108; Nova stereometria 
doliorum, 108, 110, 111, 119; Astronomia 
nova, 109; theory of maxima and mini- 
ma, 110; static approach to calculus, 111; 
forces molding thought of, 112; view of 
infinity, 117 

Kronecker, Leopold, opposed work of 
Dedekind and Cantor, 306 

La Chapelle, de, Institutions de gtomitrit, 

247 
Lacroix, S. F., 253; Traite du calcul differen- 

tiel et du calcul integral, 264; Traite" 6l6- 



Index 



343 



tnentaire, 265; limit concept, 271, 272; 
concept of the integral, 278; employment 
of divergent series, 280 

Lagrange, J. L., 224, 226, 251; quoted, 165, 
252; method of derived functions, 251-55, 
259-65, 267, 270, 272, 275, 278; concept 
of convergent series, 253, 270, 280; 
Thiorie des fonctions analytiques, 260-61, 
263 

Landen, John, The Residual Analysis, 236 

Laplace, Pierre Simon, Marquis de, 224 

Latitude of forms, see Forms, latitude of 

Law of continuity, see Continuity 

Law of uniformly difform variation, see 
Uniform acceleration 

Lebesque, integral of, 280 

Leibniz, Gottfried Wilhelm von, 4, 8, 10, 
13, 29, 47, 48, 59, 67, 82, 145, 153, 162, 
187-223, 230, 237, 287, 299, 300; infini- 
tesimal method, 28, 158, 210, 213, 219, 
243, 265, 273, 274; reference to Calcula- 
tor, 88; sought basis of calculus in genera- 
tion of magnitudes, 94; differential 
method, 122, 178, 219, 239, 244, 247, 275, 
294; views influenced by Pascal, 150; 
concept of integral as totality, 173; dif- 
ferential triangle, 182, 203; development 
of algorithmic procedure, 185, 188, 202; 
definition of the integral, 206, 278; "A 
New Method for Maxima and Minima," 
207; definition of first-order differentials, 
210; Historia et origo calculi differ entialis, 
215; quoted, 217, 218; Theodicie, 218; 
view of the infinite, 219, 297; priority 
claims of Newton and, 188, 221, 222, 246; 
law of continuity (see under Continuity) 

Leibniz's Rule, 252 

Length, concept of, 19, 31, 58 f. 

Leonardo da Vinci, 2, 88; influenced by 
Scholastic thought, 92; center of gravity 
of tetrahedron, 93; use of doctrine of 
instantaneous velocity, 130; idea of 
impetus familiar to, 177 

Leonardo of Pisa, 65, 71, 97; Liber abaci, 64 

Leucippus, 66 

Lever, law of the, 49 

L'Hospital, Marquis de, 226; Analyse des 
infinimenls petits, 238, 241 

L'Huilier, Simon, Exposition ilimenlaire des 
principes des calctds superieurs, 255; 
definition of differential quotient, 255; 
limit concept, 257, 265, 271, 272; view of 
differentials, 275; view of the integral, 278 



Limit, 15, 25, 34, 37, 50, 51, 52, 53, 62, 109, 
121, 126, 158, 174, 180, 197, 198, 200, 201, 
211, 213, 214, 218, 220, 226, 255, 260; 
definition, 7, 8, 36, 272, 287; derivative 
based on the idea of, 7, 8, 279, 293, 294; 
concept of, 24, 27, 30, 32, 43, 53 ff., 58, 
73, 77, 79, 85, 87, 91, 98, 102 ff., 106, 114, 
117, 123, 124, 132, 134, 137, 142, 153, 156, 
167, 169, 173, 176, 179, 180, 181, 195, 198, 
216, 218, 222, 228, 233, 235, 244ff., 262, 
266, 271 ff., 281, 282, 284, 286 ff., 295, 298; 
method of, lOOff., 117, 146, 150, 157, 229, 
240, 263, 264, 267, 284; doctrine of, 140; 
controversy between Robins and Jurin, 
230; calculus interpreted in terms of, 
247 ff., 254 ff., 267 ff.; involved definition 
of number, 290, 296 

Lines, incommensurability of, 19, 20; as 
velocitiesor moments, 81 ff., 114, 115, 117 

Lucretius, 68 



Mach, Ernst, view of continuity, 291, 294; 

geometric concepts, 307 
Maclaurin, Colin, 82, 234-35; concepts of 

time and instantaneous velocity, 233; 

Treatise of Fluxions, 233, 260 
Major, John, 85 
Marsilius of Inghen, 84 
Mathematical atomism, see Atomism, 

mathematical 
Mathematical formalism, 3, 67, 209, 243, 

246, 253, 272, 283, 284, 286, 305 
Mathematical Gazette, 276 
Mathematische Annalen, 289 
Maurolycus, Franciscus, 159 
Maxima and minima, 16, 85, 110, 155-59, 

163, 164, 185, 192, 208 
Mengenlehre, 296, 297; see also Infinite 

aggregates 
Meray, Charles, Nouveau precis d'analyse 

infinitesimale, 288, 289; "Remarques sur 

la nature des quantitfe d6finies," 289 
Mersenne, Marin, 165, 166 
Middle Ages, 2, 60, 61-95; speculations on 

infinite, infinitesimal, and continuity, 94; 

reaction against work of the, 96 
Miscellanea Taurinensia, 254 
Moments, 150, 178, 191, 193, 194, 195, 198, 

212, 221, 236; lines as, 114, 115, 117; 

Newton's use of, 122, 229, 235; confusion 

between fluxions and, 224, 232 
Monad, 23, 28 



344 



Index 



Montucla, Etienne, 151 

Motion, nature of, 6-7, 42, 294-95; in- 
stantaneous direction of, 57, 130 ff., 146- 
47, 166-67; uniformly accelerated, 82 ff., 
113 ff., 130, 165; graphical representation 
of, 80 ff. 



Napier, John, 122 

Neil, William, rectification of curves, 162 

Neoplatonism, 45, 177 

Newton, Sir Isaac, 4, 10, 13, 28, 29, 47, 48, 
51, 59, 82, 116, 150, 162, 174, 180, 187- 
223, 236, 268, 277, 278, 299, 300; method 
of fluxions, 58, 79, 82, 122, 129, 133, 178, 
190, 213, 220, 221, 225, 229, 278, 294; 
view of moments, 122, 235; concept of 
limit, 145, 271; invention of algorithmic 
procedures, 185, 188; concept of number, 

190, 242; De analysi, 190, 194, 200, 202, 
228, 235; use of infinite series, 190; use of 
infinitely small, and binomial theorem, 

191, 194; Methodus fluxioniim, 193, 194, 
195, 196, 200, 206, 246; definition of 
fluent and fluxion, 194, 206; De quadra- 
ture, 195, 196, 197, 200, 201, 202, 205, 
206, 221, 222, 225, 228, 247; Principia, 
197, 198, 199, 200, 201, 205, 220, 222, 225, 
228, 229, 271; concept of infinitesimal, 
226, 274; renunciation of infinitesimals, 
213, 222;"priority claims of Leibniz and, 

221, 222, 246; prime and ultimate ratios, 

222, 225, 229 ff., 254, 259, 287, 294, 300; 
Berkeley's criticism of, 224 ff.; need for 
clarification of terms used by, 224, 228, 
230; view of continuity of time, 194, 295 

Nicholas of Cusa, 2, 88, 89 ff., 93, 107, 108, 
135; quadrature of the circle, 91; views 
on infinite and infinitesimal, 92, 93; idea 
of continuity, 110; influence, 113; idea of 
impetus familiar to, 177; influenced by 
Platonic view, 88-90, 107, 304 

Nicomachus, 143-44 

Nicomedes, 71 

Nieuwentijdt, Bernard, 213, 224, 248 

Nondifferentiable continuous function, 269, 
270, 276, 282, 285, 288 

Notations of the calculus, 191, 193-95, 198, 
205, 220, 223, 252, 265, 268, 275 

Number, concept of, 15, 29, 30, 31, 36, 41, 
42, 46, 51, 52, 60, 61, 62, 63, 71, 97, 154, 
174, 176, 178, 179, 180, 190, 216, 221, 
226, 231, 242, 256, 270, 273, 281, 282, 284, 



285, 286, 289-94, 296, 298, 304, 305, 306; 
Pythagorean view, 18 ff.; Greek view, 20, 
40-45; Plato's view, 27 
Numerals, Hindu-Arabic, 63, 71 



Oldenburg, Henry, 196 

Oresme, Nicole, 65, 79 ff., 84, 96, 177, 299; 
Tractatus de latitudinibus formarum, 80, 
85, 88; Tractatus de figuratione potentia- 
rum et mensurarum, 81 ; idea of instan- 
taneous velocity, 82 ff.; consideration of 
infinite series, 86; study of maxima and 
minima, 85-86, 111, 156; concept of 
uniform acceleration, 82-83, 113, 130; 
use of geometric representations, 80 ff., 
125, 154; influence in development of 
analytic geometry, 81 ff., 154, 165, 187; 
infinitesimal concept, 84 ff., 114 ff., 193 

Oughtred, William, 170 

Oxford school, 74, 80, 84, 87 



Pacioli, Luca, 88; Sumtna de arithmetica, 64, 
97 

Pappus of Alexandria, 155; theorem, 60, 63, 
108, 139 

Parabola, quadrature of the, 49-53, 159 ff. 

Paracelsus, 177 

Parallelogram of velocities, 57, 129-34, 146 

Paris School, 80, 84, 87 

Parmenides, 23 

Pascal, Blaise, 56, 140, 142, 150, 160, 162, 
168, 209; concept of infinitesimal, 121, 
147, 148, 150, 151, 239; Potestatum 
numericarum sumtna, 148; TraitS des 
sinus du quart de cercle, 151, 153, 203; 
avoided the use of algebra and analytic 
geometry, 152-53, 302 

Pascal, Etienne, 147 

Pascal's triangle, 148, 149, 204 

Pericles, 26 

Petrus Hispanus, see John XXI, pope 

Physis, 21 

Plato, 1, 21, 25, 26, 29, 30, 37, 39, 45, 46, 48, 
89, 90, 91, 93, 291, 299, 303; concept of 
mathematics, 1, 26 ff., 60, 303, 304; 
criterion of reality, 28; influence, 89-90, 
93, 107 

Plutarch, 22; quoted, 23 

Poincare, Henri, 13 

Poisson, S. D., Traiti de mecanique, 283 

Positivism, 5, 307 



Index 



345 



Prime and ultimate ratios, 116, 195 ff., 213, 
216ff., 221, 250, 256, 262, 267; probable 
origin of term "ultimate ratio," 197; 
Newton's method, 222, 225, 229 ff., 254, 
259, 287, 294, 300; see also Number, 
concept of 

Problemum austriacum, see Gregory of St. 
Vincent, Opus geometricum 

Proclus, 20, 46; quoted, 17 

Proportion, concept of, 30 ff., 47 

Ptolemy, 59 

Pythagoras, 17, 18, 89, 299, 303; debt to 
Hindus, 19 

Pythagoreans, 1, 18 ff., 26, 39; mathemati- 
cal concepts, 17 ff., 18, 40, 176; applica- 
tion of areas, 18 ff., 32, 62, 96; problem of 
the incommensurable, 18 ff ., 61 ; influence, 
89, 93, 107, 115, 143 

Quadratures, 49-53, 91, 104f., 108, 119ff., 
124 ff., 144 ff., 149f., 159 ff., 170 ff., 191 ff., 
Quartic, solved, 97 

Ramus, Petrus, 96 

Raphson, Joseph, The History of Fluxions, 
222 

Ratios, see Number concept; Prime and 
ultimate ratios 

Rectification of curves, 133, 162, 163 

Regiomontanus, 92 

Riemann, G. F. B., 123 

Roberval, Giles Persone de, 56, 58, 125, 133, 
140ff., 151, 152, 160, 162, 168, 177, 209, 
304; concept of infinitesimal, 121, 141 ff.; 
Traits des indivisibles, 141, 147, 152; 
method of indivisibles, 141 ff., 150; 
association of numbers and geometrical 
magnitudes, 142; obscured limit idea 
through notion of indivisibles, 144; 
arithmetizing tendency in quadratures, 
142 ff., 168; method of tangents, 146 

Robervallian lines, 145, 175 

Robins, Benjamin, controversy with Berke- 
ley and Jurin, 229ff., 246, 263; A Dis- 
course Concerning . . . Newton's Methods, 
229; limit concept; 230 ff., 271; dis- 
avowal of infinitesimals, 230; criticism of 
Euler, 246; interpretation of Newton's 
prime and ultimate ratio, 247 

Rolle, Michel, 241 

Royal Society, 221 

Russell, Bertrand, 3n, 67; definition of real 
number, 293 



Scholastics, 2, 4, 26, 40, 60, 61-95; view of 
impetus, 43, 72, 177; discussions of the 
infinitely large and small, 66 ff., 77, 90; 
categorematic and syncategorematic in- 
finities, 68ff., 115, 170, 296-97; interest 
in dynamics, 72 f., 82 ff., 133; concept of 
instantaneous velocity, 73, 82; considera- 
tion of infinite series, 76 ff., 86 ff., 190; 
geometrical representation of variables, 
80 ff., Ill; knowledge of uniformly 
accelerated motion, 82 ff . ; reaction against 
methodology of, 96; influence upon 
Galileo, 112 

Sensory perception, limitations, 6, 37, 38, 
39, 43, 295 

Servois, F. J., Essai sur un nouveau mode 
^exposition des principes du calcul dif- 
ferentiel, 263 

Simplicius, 22 

Simpson, Thomas, view on instantaneous 
velocities, 234 

Skeptics, Greek, 46, 214 

Sluse, Ren6 Frangois de, 162, 185 

Snell, Willebrord, 109; Tiphys Batavus, 
152 

Sniadecki, J. B., 261 

Soto, Dominic, 85 

Spiral, of Archimedes, 55 ff., 133 

Stade, 295; argument in the, 24n; argument 
refuted by Aristotle, 44 

Stevin, Simon, 91, 104, 108, 123, 130, 145, 
152, 299; propositions on centers of 
gravity, 99 ff.; limit idea, 102, 137, 169; 
influence, 141 

Stifel, Michael, 93, 97, 108 

Strato of Lampsacus, 39 

Suiseth, Richard (Calculator), 83, 88, 96, 
113, 115, 130, 177, 299; Liber calcula- 
tionum, 69, 74, 87, 88; concept of varia- 
bility, 74 ff., consideration of infinite 
series, 76 ff., 86; use of words fluxion and 
fluent, 79, 194 

Symbols, for quantities in algebra, 98; used 
to represent infinity, 170, 297; used in 
calculus, 194, 205, 253 



Tacquet, Andreas, 138; Cylindricorum et 
annularium, 139, 151; Arithmeticae theo- 
ria et praxis, 140; method of indivisibles, 
140, 151, 181; limit concept, 140, 152, 
169, 230; propositions on geometric 
progressions, 140, 174, 190 



346 



Index 



Tangent, Greek concepts, 56 ff.; Archi- 
medes' method, 57 f.; Torricelli's method, 
128-33, 146, 157; definition, 129, 132, 

166, 174, 249; Roberval's method, 146- 
47; Fermat's method, 157-58, 162-64, 

167, 175, 183, 192-93, 202, 203; method 
of Descartes, 166; Barrow's method, 182- 
86, 192-93 

Tartaglia, Nicolo, 94, 97 

Taylor, Brook, 235, 244-45; Methodus 
incrementorum, 234 

Taylor's series, 261, 267 

Thales, 1, 16, 17, 28 

Theophrastus, 39 

Thomas, Alvarus, 85 ; Liber de triplici motu, 
87 

Thomson, William, see Kelvin, William 
Thomson, baron 

Time, concept of, 180, 194, 233, 295 

Todhunter, Isaac, 306 

Torricelli, Evangelista, 56, 58, 59, 122, 
123 ff., 134, 145, 160, 162, 168, 177, 299, 
304; view of infinitesimals, 121; De 
dimensione parabolae, 124; use of method 
of indivisibles, 125, 133, 134; De infinitis 
hyperbolis, 127, 152; method of tangents, 
128 ff., 146, 157; method of exhaustion, 
128, 133; concepts of motion and time, 
130 ff., 168, 177, 180; differential triangle, 
203 

Transfinite number, 298 

Triangle, arithmetic, see Pascal's Triangle 

Triangle, differential, 152, 163, 182, 203, 
218, 242; harmonic, 204 



Ultimate ratio, see Prime and ultimate ratio 
Uniform acceleration, 82 ff., 113ff., 130, 165 
Uniform intensity, concept of, 83 
Uniformly difform variation, law of, see 
Uniform acceleration 



122, 154ff., 167, 176 ff., 180, 193 ff., 273, 

286, 288, 294, 301; see also Graphical 

representation of variables; and Forms, 

latitude of 
Variation, law for uniformly difform, see 

Uniform acceleration 
Varignon, Pierre, 241, 280 
Velocities, parallelogram of, 57, 129-34, 

146; see also Instantaneous velocity 
Venatorius, 94 
Viete, Francois, 93, 98, 108; influence in 

development of analytic geometry, 154, 

187 
Volume, concept of, 31, 34, 58, 123 



Wallis, John, 56, 59; treatment of infinite 
and infinitesimal, 121, 170, 239, 242; 
Arithmetica infinitorum, 140, 170, 171, 
176, 189; applied analytic geometry to 
quadratures, 168f.; limit concept, 169; 
De sectionibus conicis tractatus, 170, 171; 
concept of definite integral, 173; interest 
in angle of contact, 173; arithmetization 
criticized, 179; Algebra, 201 

Weierstrass, Karl, 31; continuous nondif- 
ferentiable function, 270, 282, 288; 
trend toward formalism, 283; limit con- 
cept, 284, 286-88, 300; concept of 
irrational number, 285-300 passim; con- 
cept of the infinite, 288, 296 

Whewell, William, 256 

William of Hentisbery, see Hentisbery, 
William 

William of Occam, 66, 67, 69, 300 

Wolff, Christian, 238; concept of infinitesi- 
mal, 240; number concept, 243 

Wren, Sir Christopher, 162, 169 

Wronski, Hoen6, differential method, 261; 
Refutation de la Morie des fonctions 
analytiques, 261n, 264; limit concept, 262 



Valerio, Luca, 104 ff., 108, 112, 123, 299; 
De centro gravitatis solidorum, 104; 
proposition on inscribed and circum- 
scribed figures, 105, 124, 128, 137; limit 
concept, 106, 230 

Variability, concept of, 4, 24, 25, 42 ff., 47, 
51, 55, 56, 60, 71 ff., 78 ff., 92, 94, 96, 98, 



Xenocrates, 39 
Xenophanes of Colophon, 23 



Zeno, 43, 267, 272, 299, 300; paradoxes, 4, 
8, 23 ff., 37, 44, 53, 62, 77, 116, 138, 140, 
179, 231, 281, 287, 288, 295, 302 



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systems, art, literature, technology, mathematics, the sciences, medicine, religion, 
jurisprudence, etc. Evaluations of the contributions of scores of great men. 1964 
edition, revised and edited by scholars in the many fields represented. Total of xxix 
-f 1381pp. 21275-0, 21276-9, 21277-7 Three volumes, Paperbound $7.75 



CATALOGUE OF DOVER BOOKS 

The Philosophy of the Upanishads, Paul Deussen. Clear, detailed statement 
of upanishadic system of thought, generally considered among best available. His- 
tory of these works, full exposition of system emergent from them, parallel concepts 
in the West. Translated by A. S. Geden. xiv + 429pp. 

21616-0 Paperbound $3.00 

Language, Truth and Logic, Alfred J. Ayer. Famous, remarkably clear introduc- 
tion to the Vienna and Cambridge schools of Logical Positivism ; function of philoso- 
phy, elimination of metaphysical thought, nature of analysis, similar topics. "Wish 
I had written it myself," Bertrand Russell. 2nd, 1946 edition. 160pp. 

20010-8 Paperbound $1.35 

The Guide for the Perplexed, Moses Maimonides. Great classic of medieval 
Judaism, major attempt to reconcile revealed religion (Pentateuch, commentaries) 
and Aristotelian philosophy. Enormously important in all Western thought. Un- 
abridged Friedlander translation. 50-page introduction, lix + 4l4pp. 

(USO) 20351-4 Paperbound $2.50 

Occult and Supernatural Phenomena, D. H. Rawcliffe. Full, serious study 
of the most persistent delusions of mankind: crystal gazing, mediumistic trance, 
stigmata, lycanthropy, fire walking, dowsing, telepathy, ghosts, ESP, etc., and their 
relation to common forms of abnormal psychology. Formerly Illusions and Delusions 
of the Supernatural and the Occult, iii -f 551pp. 20503-7 Paperbound $3.50 



The Egyptian Book of the Dead: The Papyrus of Ani, E. A. Wallis Budge. 
Full hieroglyphic text, interlinear transliteration of sounds, word for word trans- 
lation, then smooth, connected translation; Theban recension. Basic work in Ancient 
Egyptian civilization ; now even more significant than ever for historical importance, 
dilation of consciousness, etc. clvi + 377pp. 6V2 * 9 l /+. 

21866-X Paperbound $3.95 

Psychology of Music, Carl E. Seashore. Basic, thorough survey of everything 
known about psychology of music up to 1940's; essential reading for psychologists, 
musicologists. Physical acoustics; auditory apparatus; relationship of physical 
sound to perceived sound ; role of the mind in sorting, altering, suppressing, creating 
sound sensations; musical learning, testing for ability, absolute pitch, other topics. 
Records of Caruso, Menuhin analyzed. 88 figures, xix + 408pp. 

21851-1 Paperbound $2.75 

The I Ching (The Book of Changes), translated by James Legge. Complete 
translated text plus appendices by Confucius, of perhaps the most penetrating divina- 
tion book ever compiled. Indispensable to all study of early Oriental civilizations. 
3 plates, xxiii + 448pp. 21062-6 Paperbound $3.00 

The Upanishads, translated by Max Miiller. Twelve classical upanishads: Chan- 
dogya, Kena, Aitareya, Kaushitaki, Isa, Katha, Mundaka, Taittiriyaka, Brhadaran- 
yaka, Svetasvatara, Prasna, Maitriyana. 160-page introduction, analysis by Prof. 
Miiller. Total of 826pp. 20398-0, 20399-9 Two volumes, Paperbound $5.00 



CATALOGUE OF DOVER BOOKS 

Algebras and Their Arithmetics, Leonard E. Dickson. Complete background 
for advanced study of abstract algebra. Clear rigorous exposition of the structures 
of many special algebras, from an elementary introduction to linear transformations, 
matrices and complex numbers to a generalization of the classic theory of integral 
algebraic numbers. Each definition and theorem illustrated by a simple example. 
xii + 241pp. 60616-3 Paperbound $1.50 

Astronomy and Cosmogony, Sir James Jeans. Modern classic of exposition, 
Jean's latest work. Descriptive astronomy, atrophysics, stellar dynamics, cosmology, 
presented on intermediate level. 16 illustrations. Preface by Lloyd Motz. xv + 
428pp. 60923-5 Paperbound $3.50 

Experimental Spectroscopy, Ralph A. Sawyer. Discussion of techniques and 
principles of prism and grating spectrographs used in research. Full treatment of 
apparatus, construction, mounting, photographic process, spectrochemical analysis, 
theory. Mathematics kept to a minimum. Revised (1961) edition. 110 illustra- 
tions, x -f- 358pp. 61045-4 Paperbound $3.00 

Theory of Light, Richard von Mises. Introduction to fluid dynamics, explaining 
fully the physical phenomena and mathematical concepts of aeronautical engineer- 
ing, general theory of stability, dynamics of incompressible fluids and wing theory. 
Still widely recommended for clarity, though limited to situations in which air 
compressibility effects are unimportant. New introduction by K. H. Hohenemser. 
408 figures, xvi + 629pp. 60541-8 Paperbound $3.75 

Airplane Structural Analysis and Design, Ernest E. Sechler and Louis G. 
Dunn. Valuable source work to the aircraft and missile designer: applied and 
design loads, stress-strain, frame analysis, plates under normal pressure, engine 
mounts, landing gears, etc. 47 problems. 256 figures, xi -\- 420pp. 

61043-8 Paperbound $2.50 

Photoelasticity: Principles and Methods, H. T. Jessop and F. C. Harris. 
An introduction to general and modern developments in 2- and 3 -dimensional stress 
analysis techniques. More advanced mathematical treatment given in appendices. 
164 figures, viii + 184pp. 6y 8 x 9^4- (USO) 60720-8 Paperbound $2.00 



The Measurement of Power Spectra From the Point of View of Com- 
munications Engineering, Ralph B. Blackman and John W. Tukey. Techniques 
for measuring the power spectrum using elementary transmission theory and theory 
of statistical estimation. Methods of acquiring sound data, procedures for reducing 
data to meaningful estimates, ways of interpreting estimates. 36 figures and tables. 
Index, x + 190pp. 60507-8 Paperbound $2.50 



Gaseous Conductors: Theory and Engineering Applications, James D. 
Cobine. An indispensable reference for radio engineers, physicists and lighting 
engineers. Physical backgrounds, theory of space charges, applications in circuit 
interrupters, rectifiers, oscillographs, etc. 83 problems. Over 600 figures, xx + 
606pp. 60442-X Paperbound $3.75 



CATALOGUE OF DOVER BOOKS 

Adventures of an African Slaver, Theodore Canot. Edited by Brantz Mayer. 
A detailed portrayal of slavery and the slave trade, 1820-1840. Canot, an established 
trader along the African coast, describes the slave economy of the African kingdoms, 
the treatment of captured negroes, the extensive journeys in the interior to gather 
slaves, slave revolts and their suppression, harems, bribes, and much more. Full and 
unabridged republication of 1854 edition. Introduction by Malcom Cowley. 16 
illustrations, xvii + 448pp. 22456-2 Paperbound $3.50 

My Bondage and My Freedom, Frederick Douglass. Born and brought up in 
slavery, Douglass witnessed its horrors and experienced its cruelties, but went on 
to become one of the most outspoken forces in the American anti-slavery movement. 
Considered the best of his autobiographies, this book graphically describes the in- 
human treatment of slaves, its effects on slave owners and slave families, and how 
Douglass's determination led him to a new life. Unaltered reprint of 1st (1855) 
edition, xxxii + 464pp. 22457-0 Paperbound $2.50 

The Indians' Book, recorded and edited by Natalie Curtis. Lore, music, narratives, 
dozens of drawings by Indians themselves from an authoritative and important 
survey of native culture among Plains, Southwestern, Lake and Pueblo Indians. 
Standard work in popular ethnomusicology. 149 songs in full notation. 23 draw- 
ings, 23 photos, xxxi + 584pp. 6% x 9%. 21939-9 Paperbound $4.50 



Dictionary of American Portraits, edited by Hayward and Blanche Cirker. 
4024 portraits of 4000 most important Americans, colonial days to 1905 (with a 
few important categories, like Presidents, to present) . Pioneers, explorers, colonial 
figures, U. S. officials, politicians, writers, military and naval men, scientists, inven- 
tors, manufacturers, jurists, actors, historians, educators, notorious figures, Indian 
chiefs, etc. All authentic contemporary likenesses. The only work of its kind in 
existence; supplements all biographical sources for libraries. Indispensable to any- 
one working with American history. 8,000-item classified index, finding lists, other 
aids, xiv + 756pp. 9\/ 4 x 12%. 21823-6 Clothbound $30.00 



Tritton's Guide to Better Wine and Beer Making for Beginners, S. M. 
Tritton. All you need to know to make family-sized quantities of over 100 types 
of grape, fruit, herb and vegetable wines ; as well as beers, mead, cider, etc. Com- 
plete recipes, advice as to equipment, procedures such as fermenting, bottling, and 
storing wines. Recipes given in British, U. S., and metric measures. Accompanying 
booklet lists sources in U. S. A. where ingredients may be bought, and additional 
information. 11 illustrations. 157pp. 5% x 8Vs- 

(USO) 22090-7 Clothbound $3.50 

Gardening With Herbs for Flavor and Fragrance, Helen M. Fox. How to 
grow herbs in your own garden, how to use them in your cooking (over 55 recipes 
included), legends and myths associated with each species, uses in medicine, per- 
fumes, etc. — these are elements of one of the few books written especially for Amer- 
ican herb fanciers. Guides you step-by-step from soil preparation to harvesting and 
storage for each type of herb. 12 drawings by Louise Mansfield, xiv + 334pp. 

22540-2 Paperbound $2.50 



CATALOGUE OF DOVER BOOKS 

American Food and Game Fishes, David S. Jordan and Barton W. Evermann. 
Definitive source of information, detailed and accurate enough to enable the sports- 
man and nature lover to identify conclusively some 1,000 species and sub-species 
of North American fish, sought for food or sport. Coverage of range, physiology, 
habits, life history, food value. Best methods of capture, interest to the angler, advice 
on bait, fly-fishing, etc. 338 drawings and photographs. 1 + 574pp. 6% xV/fc. 

22383-1 Paperbound $4.50 

The Frog Book, Mary C. Dickerson. Complete with extensive finding keys, over 
300 photographs, and an introduction to the general biology of frogs and toads, this 
is the classic non-technical study of Northeastern and Central species. 58 species; 
290 photographs and 16 color plates, xvii + 253pp. 

21973-9 Paperbound $4.00 

The Moth Book: A Guide to the Moths of North America, William J. 
Holland. Classical study, eagerly sought after and used for the past 60 years. Clear 
identification manual to more than 2,000 different moths, largest manual in existence. 
General information about moths, capturing, mounting, classifying, etc., followed by 
species by species descriptions. 263 illustrations plus 48 color plates show almost 
every species, full size. 1968 edition, preface, nomenclature changes by A. E. 
Brower. xxiv + 479pp. of text. 6y 2 x 9%. 

21948-8 Paperbound $5.00 

The Sea-Beach at Ebb-Tide, Augusta Foote Arnold. Interested amateur can iden- 
tify hundreds of marine plants and animals on coasts of North America; marine 
algae; seaweeds; squids; hermit crabs; horse shoe crabs; shrimps; corals; sea anem- 
ones; etc. Species descriptions cover: structure; food; reproductive cycle; size; 
shape; color; habitat; etc. Over 600 drawings. 85 plates, xii -f 490pp. 

21949-6 Paperbound $3.50 

Common Bird Songs, Donald J. Borror. 33 x /3 12-inch record presents songs of 60 
important birds of the eastern United States. A thorough, serious record which pro- 
vides several examples for each bird, showing different types of song, individual 
variations, etc. Inestimable identification aid for birdwatcher. 32-page booklet gives 
text about birds and songs, with illustration for each bird. 

21829-5 Record, book, album. Monaural. $2.75 

Fads and Fallacies in the Name of Science, Martin Gardner. Fair, witty ap- 
praisal of cranks and quacks of science: Atlantis, Lemuria, hollow earth, flat earth, 
Velikovsky, orgone energy, Dianetics, flying saucers, Bridey Murphy, food fads, 
medical fads, perpetual motion, etc. Formerly "In the Name of Science." x + 363pp. 

20394-8 Paperbound $2.00 

Hoaxes, Curtis D. MacDougall. Exhaustive, unbelievably rich account of great 
hoaxes: Locke's moon hoax, Shakespearean forgeries, sea serpents, Loch Nesj 
monster, Cardiff giant, John Wilkes Booth's mummy, Disumbrationist school of art, 
dozens more; also journalism, psychology of hoaxing. 54 illustrations, xi + 338pp. 

20465-0 Paperbound $2.75 



CATALOGUE OF DOVER BOOKS 

Advanced Calculus, Edwin B. Wilson. Widely regarded as among the most 
useful and comprehensive texts in this subject. Many chapters, such as those on 
vector functions, ordinary differential equations, special functions, calculus of 
variations, elliptic functions and partial differential equations, are excellent intro- 
ductions to their branches of higher mathematics. More than 1300 exercises speed 
comprehension and indicate applications, ix -f 566pp. 

60504-3 Paperbound $3.00 

A Treatise on Advanced Calculus, Philip Franklin. Comprehensive, logical 
treatment of theory of calculus and allied subjects. Provides solid basis for gradu- 
ate study without going as far as texts on real variable theory. Theory stressed over 
applications and techniques. 612 exercise problems with solution hints. 28 figures. 
xi -f 595pp. 61252-X Paperbound $4.00 

Hydrodynamics, Sir Horace Lamb. Standard reference and study work, almost 
inexhaustible in coverage of classical material. Unexcelled for fundamental theo- 
rems, equations, detailed methods of solution: equations of motion, integration 
of equations, irrotational motion, motion oi liquid in two dimensions, motion of 
solids through liquids, vortex motion, tidal waves, waves of expansion, surface 
waves, viscosity, rotating liquids, etc. 6th enlarged edition. 119 figures, xv + 
738pp. 6x9. (USO) 60256-7 Paperbound $4.00 

Electromagnetism, John C. Slater and Nathaniel H. Frank. Introductory study 
by leading men in the field supplies basic material on electrostatics and magneto- 
statics, then concentrates on electromagnetic theory, ranging over many areas and 
touching on electrical engineering. Also covers equations and theorems of Gauss, 
Poisson, Laplace and Green, dielectrics, magnetic fields of linear and circular 
currents, electromagnetic induction and Maxwell's equations, wave guides and 
cavity resonators, Huygens' principle, etc. A knowledge of calculus and differential 
equations required. Problems are supplied. 39 figures, xii + 240pp. 

62263-0 Paperbound $2.75 

Applied Hydro- and Aeromechanics, Ludwig Prandtl, O. G. Tietjens. Methods 
valuable to engineers: flow in pipes, boundary layers, airfoil theory, entry condi- 
tions, turbulent flow in pipes, drag, etc. 226 figures, 287 photographic plates, 
xvi + 311pp. 60375-X Paperbound $2.50 

Basic Optics and Optical Instruments, U. S. Navy. Navy elementary train- 
ing manual, clearly treating the composition of optical glass, characteristics of light, 
elements of mirrors, prisms and lenses, construction of optical instruments, main- 
tenance and repair procedures. Formerly titled Opticalman 2 & 3. Nearly 600 
charts, diagrams, photgraphs and drawings, vi -f- 485pp. 6Y 2 x 9Y 4 . 

22291-8 Paperbound $3.50 

Mechanics of the Gyroscope: The Dynamics of Rotation, Richard F. 
Diemel. Applications of gyroscopic phenomena stressed in this elementary treat- 
ment of the dynamics of rotation. Covers velocity on a moving curve, gyroscopic 
phenomena and apparatus, the gyro-compass, stabilizers (ships and monorail ve- 
hicles). "Remarkably concise and generous treatment," Industrial Laboratories. 
75 figures. 136 exercises, ix + 192pp. 60066-1 Paperbound $1.75 



CATALOGUE OF DOVER BOOKS 

Planets, Stars and Galaxies: Descriptive Astronomy for Beginners, A. E. 
Fanning. Comprehensive introductory survey of astronomy: the sun, solar system, 
stars, galaxies, universe, cosmology; up-to-date, including quasars, radio stars, etc. 
Preface by Prof. Donald Menzel. 24pp. of photographs. 189pp. 5*4 x 8*4. ' 

21680-2 Paperbound $1.50 

Teach Yourself Calculus, P. Abbott. With a good background in algebra and 
trig, you can teach yourself calculus with this book. Simple, straightforward intro- 
duction to functions of all kinds, integration, differentiation, series, etc. "Students 
who are beginning to study calculus method will derive great help from this book." 
Faraday House Journal. 308pp. 20683-1 Clothbound $2.00 

Teach Yourself Trigonometry, P. Abbott. Geometrical foundations, indices and 
logarithms, ratios, angles, circular measure, etc. are presented in this sound, easy-to- 
use text. Excellent for the beginner or as a brush up, this text carries the student 
through the solution of triangles. 204pp. 20682-3 Clothbound $2.00 

Teach Yourself Anatomy, David LeVay. Accurate, inclusive, profusely illus- 
trated account of structure, skeleton, abdomen, muscles, nervous system, glands, 
brain, reproductive organs, evolution. "Quite the best and most readable account,' 
Medical Officer. 12 color plates. 164 figures. 311pp. 4% x 7. 

21651-9 Clothbound $2.50 

Teach Yourself Physiology, David LeVay. Anatomical, biochemical bases ; di- 
gestive, nervous, endocrine systems; metabolism; respiration; muscle; excretion; 
temperature control; reproduction. "Good elementary exposition," The Lancet. 6 
color plates. 44 illustrations. 208pp. 4y 4 x 7. 21658-6 Clothbound $2.50 

The Friendly Stars, Martha Evans Martin. Classic has taught naked-eye observa- 
tion of stars, planets to hundreds of thousands, still not surpassed for charm, lucidity, 
adequacy. Completely updated by Professor Donald H. Menzel, Harvard Observa- 
tory. 25 illustrations. 16 x 30 chart, x + 147pp. 21099-5 Paperbound $1.25 

Music of the Spheres: The Material Universe from Atom to Quasar, 
Simply Explained, Guy Murchie. Extremely broad, brilliantly written popular 
account begins with the solar system and reaches to dividing line between matter and 
nonmatter; latest understandings presented with exceptional clarity. Volume One: 
Planets, stars, galaxies, cosmology, geology, celestial mechanics, latest astronomical 
discoveries; Volume Two: Matter, atoms, waves, radiation, relativity, chemical 
action, heat, nuclear energy, quantum theory, music, light, color, probability, anti- 
matter, antigravity, and similar topics. 319 figures. 1967 (second) edition. Total 
of xx + 644pp. 21809-0, 21810-4 Two volumes, Paperbound $5.00 

Old-Time Schools and School Books, Clifton Johnson. Illustrations and rhymes 
from early primers, abundant quotations from early textbooks, many anecdotes of 
school life enliven this study of elementary schools from Puritans to middle 19th 
century. Introduction by Carl Withers. 234 illustrations, xxxiii + 381pp. 

21031-6 Paperbound $2.50 



CATALOGUE OF DOVER BOOKS 

Basic Electricity, U. S. Bureau of Naval Personel. Originally a training course, 
best non-technical coverage of basic theory of electricity and its applications. Funda- 
mental concepts, batteries, circuits, conductors and wiring techniques, AC and DC, 
inductance and capacitance, generators, motors, transformers, magnetic amplifiers, 
synchros, servomechanisms, etc. Also covers blue-prints, electrical diagrams, etc. 
Many questions, with answers. 349 illustrations, x 4- 448pp. 6V^ x 9*4. 

20973-3 Paperbound $3.00 

Tensors For Circuits, Gabriel Kron. The purpose of this volume was to develop 
a new mathematical method of analyzing engineering problems — through tensor 
analysis — which has since proven its usefulness especially in electrical and structural 
networks in computers. Introduction by Banesh Hoffmann. Formerly A Short 
Course in Tensor Analysis. Over 800 figures, xviii -}- 250pp. 

60534-5 Paperbound $2.00 

Information Theory, Stanford Goldman. A thorugh presentation of the work 
of C. E. Shannon and to a lesser extent Norbert Weiner, at a mathematical level 
understandable to first-year graduate students in electrical engineering. In addi- 
tion, the basic and general aspects of information theory are developed at an 
elementary level for workers in non-mathematical sciences. Table of logarithms to 
base 2. xiii -f 385pp. 62209-6 Paperbound $3.50 



Introduction to the Statistical Dynamics of Automatic Control 
Systems, V. V. Solodovnikov. General theory of control systems subjected to 
random signals. Theory of linear analysis, statistics of random signals, theory of 
linear prediction and filtering. For advanced and graduate-level students. Trans- 
lated by John B. Thomas and Lotfi A. Zadeh. xxi + 307pp. 

60420-9 Paperbound $3.00 

Fundamental of Hydro- and Aeromechanics, Ludwig Prandtl and O. G. 
Tietjens. Tietjens' famous expansion of Professor Prandtl's Kaiser Wilhelm 
Institute lectures. Much original material included in coverage of statics of liquids 
and gases, kinematics of liquids and gases, dynamics of non-viscous liquids. Proofs 
are rigorous and use vector analysis. Translated by L. Rosenhead. 186 figures, 
xvi + 270pp. 60374-1 Paperbound $2.25 



Mathematical Methods for Scientists and Engineers, L. P. Smith. Full 
investigation of methods, practical description of conditions where used: elements 
of real functions, differential and integral calculus, space geometry, residues, 
vectors and tensors, Bessel functions, etc. Many examples from scientific literature 
completely worked out. 368 problems for solution, 100 diagrams, x + 453pp. 

60220-6 Paperbound $2.75 

Computational Methods of Linear Algebra, V. N. Faddeeva. Only work in 
English to present classical and modern Russian computational methods of linear 
algebra, including the work of A. N. Krylov, A. M. Danilevsky, D. K. Faddeev 
and others. Detailed treatment of the derivation of numerical solutions to problems 
of linear algebra. Translated by Curtis D. Benster. 23 carefully prepared tables. 
New bibliography, x + 252pp. 60424-1 Paperbound $2.50 



CATALOGUE OF DOVER BOOKS 

Two Little Savages; Being the Adventures of Two Boys Who Lived as 
Indians and What They Learned, Ernest Thompson Seton. Great classic of 
nature and boyhood provides a vast range of woodlore in most palatable form, a 
genuinely entertaining story. Two farm boys build a teepee in woods and live in it 
for a month, working out Indian solutions to living problems, star lore, birds and 
animals, plants, etc. 293 illustrations, vii + 286pp. 

20985-7 Paperbound $2.50 

Peter Piper's Practical Principles of Plain & Perfect Pronunciation. 
Alliterative jingles and tongue-twisters of surprising charm, that made their first 
appearance in America about 1830. Republished in full with the spirited woodcut 
illustrations from this earliest American edition. 32pp. 4 l / 2 x 6 3 / 8 . 

22560-7 Paperbound $1.00 

Science Experiments and Amusements for Children, Charles Vivian. 73 easy 
experiments, requiring only materials found at home or easily available, such as 
candles, coins, steel wool, etc.; illustrate basic phenomena like vacuum, simple 
chemical reaction, etc. All safe. Modern, well-planned. Formerly Science Games 
for Children. 102 photos, numerous drawings. 96pp. 6% x 9%. 

21856-2 Paperbound $1.25 

An Introduction to Chess Moves and Tactics Simply Explained, Leonard 
Barden. Informal intermediate introduction, quite strong in explaining reasons for 
moves. Covers basic material, tactics, important openings, traps, positional play in 
middle game, end game. Attempts to isolate patterns and recurrent configurations. 
Formerly Chess. 58 figures. 102pp. (USO) 21210-6 Paperbound $1.25 



Lasker's Manual of Chess, Dr. Emanuel Lasker. Lasker was not only one of the 
five great World Champions, he was also one of the ablest expositors, theorists, and 
analysts. In many ways, his Manual, permeated with his philosophy of battle, filled 
with keen insights, is one of the greatest works ever written on chess. Filled with 
analyzed games by the great players. A single-volume library that will profit almost 
any chess player, beginner or master. 308 diagrams, xli x 349pp. 

20640-8 Paperbound $2.75 

The Master Book of Mathematical Recreations, Fred Schuh. In opinion of 
many the finest work ever prepared on mathematical puzzles, stunts, recreations ; 
exhaustively thorough explanations of mathematics involved, analysis of effects, 
citation of puzzles and games. Mathematics involved is elementary. Translated by 
F. Gobel. 194 figures, xxiv + 430pp. 22134-2 Paperbound $3.00 



Mathematics, Magic and Mystery, Martin Gardner. Puzzle editor for Scientific 
American explains mathematics behind various mystifying tricks: card tricks, stage 
"mind reading," coin and match tricks, counting out games, geometric dissections, 
etc. Probability sets, theory of numbers clearly explained. Also provides more than 
400 tricks, guaranteed to work, that you can do. 135 illustrations, xii + 176pp. 

20338-2 Paperbound $1.50 



CATALOGUE OF DOVER BOOKS 

Poems of Anne Bradstreet, edited with an introduction by Robert Hutchinson. 
A new selection of poems by America's first poet and perhaps the first significant 
woman poet in the English language. 48 poems display her development in works 
of considerable variety — love poems, domestic poems, religious meditations, formal 
elegies, "quaternions," etc. Notes, bibliography, viii + 222pp. 

22160-1 Paperbound $2.00 

Three Gothic Novels: The Castle of Otranto by Horace Walpole; 
Vathek by William Beckford; The Vampyre by John Polidori, with Frag- 
ment of a Novel by Lord Byron, edited by E. F. Bleiler. The first Gothic 
novel, by Walpole; the finest Oriental tale in English, by Beckford; powerful 
Romantic supernatural story in versions by Polidori and Byron. All extremely 
important in history of literature; all still exciting, packed with supernatural 
thrills, ghosts, haunted castles, magic, etc. xl + 291pp. 

21232-7 Paperbound $2.00 

The Best Tales of Hoffmann, E. T. A. HofTmann. 10 of Hoffmann's most 
important stories, in modern re-editings of standard translations: Nutcracker and 
the King of Mice, Signor Formica, Automata, The Sandman, Rath Krespel, The 
Golden Flowerpot, Master Martin the Cooper, The Mines of Falun, The King's 
Betrothed, A New Year's Eve Adventure. 7 illustrations by Hoffmann. Edited 
by E. F. Bleiler. xxxix + 419pp. 21793-0 Paperbound $2.50 

Ghost and Horror Stories of Ambrose Bierce, Ambrose Bierce. 23 strikingly 
modern stories of the horrors latent in the human mind: The Eyes of the Panther, 
The Damned Thing, An Occurrence at Owl Creek Bridge, An Inhabitant of Carcosa, 
etc., plus the dream-essay, Visions of the Night. Edited by E. F. Bleiler. xxii 
+ 1 99pp. 20767-6 Paperbound $1.50 

Best Ghost Stories of J. S. LeFanu, J. Sheridan LeFanu. Finest stories by 
Victorian master often considered greatest supernatural writer of all. Carmilia, 
Green Tea, The Haunted Baronet, The Familiar, and 12 others. Most never before 
available in the U. S. A. Edited by E. F. Bleiler. 8 illustrations from Victorian 
publications, xvii + 467pp. 20415-4 Paperbound $2.50 

The Time Stream, the Greatest Adventure, and the Purple Sapphire — 
Three Science Fiction Novels, John Taine (Eric Temple Bell). Great Amer- 
ican mathematician was also foremost science fiction novelist of the 1920's. The 
Time Stream, one of all-time classics, uses concepts of circular time; The Greatest 
Adventure, incredibly ancient biological experiments from Antarctica threaten to 
escape; The Purple Sapphire, superscience, lost races in Central Tibet, survivors 
of the Great Race. 4 illustrations by Frank R. Paul, v + 532pp. 

21180-0 Paperbound $3.00 

Seven Science Fiction Novels, H. G. Wells. The standard collection of the 
great novels. Complete, unabridged. First Men in the Moon, Island of Dr. Moreau, 
War of the Worlds, Food of the Gods, Invisible Man, Time Machine, In the Days 
of the Comet. Not only science fiction fans, but every educated person owes it to 
himself to read these novels. 1015pp. 20264-X Clothbound $5.00 



CATALOGUE OF DOVER BOOKS 

Tables of Functions: With Formulae and Curves, Eugene Jahnke and Fritz 
Emde. The revised fourth edition, with the 76-page appendix of tables and 
formulae of elementary functions. Sine, cosine, logarithmic integral; error integral; 
Riemann-Zeta function; Mathieu functions; cubic equations; exponential func- 
tion; the hyperbolic functions; much more. 212 figures, xii + 382pp. 

(USO) 60133-1 Paperbound $2.50 

Statistical Adjustment of Data, W. Edwards Deming. Introduction to basic 
concepts of statistics, curve fitting, least squares solution, conditions without 
parameter, conditions containing parameters. 26 exercises worked out. Some back- 
ground in differential calculus desirable. Tables, x + 26lpp. 

6123 5-X Paperbound $2.00 

The Works of Archimedes, With the Method of Archimedes, edited by 
T. L. Heath. All the known works of the great Greek mathematician. The editor's 
186-page introduction describes the relation of Archimedes to his predecessors, his 
life, and his thought. Supplement: The Method of Archimedes, recently discovered 
by Heiberg. clxxxvi + 377pp. 60009-2 Paperbound $3.50 



A Concise History of Mathematics, Dirk J. Struik. The best brief history of 
mathematics. Stresses origins and covers every major figure in math history from 
the ancient Near East to the great figures of the 19th century such as Fourier, 
Gauss, Riemann, Cantor, many others. "A tremendous undertaking," American 
Mathematical Monthly. Third revised edition. 41 illustrations, x + 195pp. 

(EUK) 60255-9 Paperbound $2.00 

An Introduction to Relaxation Methods, Frederick S. Shaw. Use of relaxa- 
tion methods for solution of differential equations, written by co-worker with Sir 
Richard Southwell, who developed relaxation methods. Deals with application of 
a general computational process which has been extended to include almost all 
branches of applied mechanics and physics. Treatment is mathematical rather than 
physical. Detailed demonstrations and examples. 253 diagrams, 72 tables. 396pp. 

60244-3 Paperbound $3-25 

Mathematical Tables and Formulas, Robert D. Carmichael and Edwin R. 
Smith. All tables necessary for college algebra and trigonometry. Five-place 
logarithms, sines and tangents, trigonometric functions, powers, roots, reciprocals, 
exponential and hyperbolic functions, formulas and theorems from geometry and 
calculus. Very rich collection, viii + 269pp. 60111-0 Paperbound $1.50 



Mathematical Tables of Elementary and Some Higher Mathematical 
Functions, Herbert B. Dwight. Almost every function of importance in applied 
mathematics, engineering and the physical sciences. Trigonometric functions and 
their inverse functions to thousandths of radians, logarithms, hyperbolic functions, 
elliptic functions of the first and second kind, over 60 pages of Bessel functions, 
Euler numbers and their logs to base ten, and many others. Over half have 
columns of difference to facilitate interpolation, viii -f- 231pp. 

60445-4 Paperbound $2.50 



CATALOGUE OF DOVER BOOKS 

Planets, Stars and Galaxies: Descriptive Astronomy for Beginners, A. E. 
Fanning. Comprehensive introductory survey of astronomy: the sun, solar system, 
stars, galaxies, universe, cosmology; up-to-date, including quasars, radio stars, etc. 
Preface by Prof. Donald Menzel. 24pp. of photographs. 189pp. 5% x 8Y 4 . 

21680-2 Paperbound $1.50 

Teach Yourself Calculus, P. Abbott. With a good background in algebra and 
trig, you can teach yourself calculus with this book. Simple, straightforward intro- 
duction to functions of all kinds, integration, differentiation, series, etc. "Students 
who are beginning to study calculus method will derive great help from this book." 
Faraday House Journal. 308pp. 20683-1 Clothbound $2.00 

Teach Yourself Trigonometry, P. Abbott. Geometrical foundations, indices and 
logarithms, ratios, angles, circular measure, etc. are presented in this sound, easy-to- 
use text. Excellent for the beginner or as a brush up, this text carries the student 
through the solution of triangles. 204pp. 20682-3 Clothbound $2.00 

Teach Yourself Anatomy, David LeVay. Accurate, inclusive, profusely illus- 
trated account of structure, skeleton, abdomen, muscles, nervous system, glands, 
brain, reproductive organs, evolution. "Quite the best and most readable account,' 
Medical Officer. 12 color plates. 164 figures. 311pp. 4y 4 x7. 

21651-9 Clothbound $2.50 

Teach Yourself Physiology, David LeVay. Anatomical, biochemical bases ; di- 
gestive, nervous, endocrine systems; metabolism; respiration; muscle; excretion; 
temperature control; reproduction. "Good elementary exposition," The Lancet. 6 
color plates. 44 illustrations. 208pp. 4y 4 x 7. 21658-6 Clothbound $2.50 

The Friendly Stars, Martha Evans Martin. Classic has taught naked-eye observa- 
tion of stars, planets to hundreds of thousands, still not surpassed for charm, lucidity, 
adequacy. Completely updated by Professor Donald H. Menzel, Harvard Observa- 
tory. 25 illustrations. 16 x 30 chart. x+ 147pp. 2*1099-5 Paperbound $1.25 

Music of the Spheres: The Material Universe from Atom to Quasar, 
Simply Explained, Guy Murchie. Extremely broad, brilliantly written popular 
account begins with the solar system and reaches to dividing line between matter and 
nonmatter; latest understandings presented with exceptional clarity. Volume One: 
Planets, stars, galaxies, cosmology, geology, celestial mechanics, latest astronomical 
discoveries; Volume Two: Matter, atoms, waves, radiation, relativity, chemical 
action, heat, nuclear energy, quantum theory, music, light, color, probability, anti- 
matter, antigravity, and similar topics. 319 figures. 1967 (second) edition. Total 
of xx + 644pp. 21809-0, 21810-4 Two volumes, Paperbound $5.00 

Old-Time Schools and School Books, Clifton Johnson. Illustrations and rhymes 
from early primers, abundant quotations from early textbooks, many anecdotes of 
school life enliven this study of elementary schools from Puritans to middle 19th 
century. Introduction by Carl Withers. 234 illustrations, xxxiii + 381pp. 

21031-6 Paperbound $2.50 



CATALOGUE OF DOVER BOOKS 

Mechanics of Materials, Alvin Sloane. Over 500 problems are used to illustrate 
the theory of elasticity introduced in this lucid text, ideal for class use. 299 
figures. 553 problems, many answered. 17 tables, xvii + 468pp. 

61767-X Paperbound $2.75 
Answer Book: Detailed, fully worked solutions for all 553 problems. 9V 2 x 12y 4 . 
26pp- 62513-3 Paperbound 75c 

Applied Optics and Optical Design, A. E. Conrady. Standard work for 
designers. Detailed step-by-step exposition, moving from fundamentals to design 
of various systems. Volume 1 covers ordinary raytracing methods, primary aberra- 
tion, telescopes, low-power microscope, photographic equipment. Volume 2 (com- 
pleted by Rudolph Kingslake) covers high-power microscope, anastigmatic photo- 
graphic objectives. 193 diagrams. Total of xiv -f 84lpp. 

60611-2, 60612-2 Two volumes, Paperbound $7.50 

Introduction to Electronics, U. S. Bureau of Naval Personnel. Basic con- 
cepts, techniques, equipment on elementary level, no background required. Power 
supplies, electron tubes in circuits, operation of transistors, servosystems, radio, 
radar, sonar, etc. 155 figures, viii + 145pp. 6y 2 x 914. 

21283-1 Paperbound $1.25 

Basic Electronics, U. S. Bureau of Naval Personnel. Using nothing more ad- 
vanced than elementary electricity and mathematics, this manual covers electron 
tubes, circuits, antennas, radar, etc. — even transistors. 430 wiring diagrams, cut- 
aways, photos, vii + 459pp. 6Y 2 x 9 l A- 21076-6 Paperbound $2.95 



Basic Theory and Application of Transistors, U. S. Department of the 
Army. Fundamental theory, applications for persons with minimal electronic 
background. Physical basis, amplifiers, bias stabilization, analysis and comparison, 
amplifiers, pulse and switching circuits, modulation, semiconductors, etc. Thorough, 
nothing left out. 240 diagrams. 263pp. 6I/2 x 9V 4 . 

20380-8 Paperbound $1.75 

Microwave Electronics, John C. Slater. Full treatment of microwave elec- 
tronics and its electromagnetic basis covers passive microwave circuits and active 
devices such as klystrons, magnetrons, etc. A detailed mathematical analysis is 
supplemented with experiments and physical pictures. Work is chiefly concerned 
with basic physical principles rather than specific devices. 91 figures, xiv + 
406pp. 62264-9 Paperbound $3.50 

Handbook of Mathematical Functions With Formulas, Graphs, and 
Mathematical Tables, edited by Milton Abramowitz and Irene A. Stegun. 
Vast compendium of tables of functions designed to meet needs of scientists in all 
fields. 29 sets of tables, some to as high as 20 places ; error function, Bessel func- 
tions, Struve functions, Coulomb wave functions, hypergeometric functions, elliptic 
integrals, Mathieu functions, orthogonal polynomials, probability functions, Laplace 
transforms, etc. Originally published by U. S. Department of Commerce. Revised 
edition, xiv + 1046pp. 8 x 10y 2 . 61272-4 Paperbound $5.00 



CATALOGUE OF DOVER BOOKS 

Johann Sebastian Bach, Philipp Spitta. One of the great classics of musicology, 
this definitive analysis of Bach's music (and life) has never been surpassed. Lucid, 
nontechnical analyses of hundreds of pieces (30 pages devoted to St. Matthew Pas- 
sion, 26 to B Minor Mass). Also includes major analysis of 18th-century music. 
450 musical examples. 40-page musical supplement. Total of xx + 1799pp. 

(EUK) 22278-0, 22279-9 Two volumes, Clothbound $15.00 

Mozart and His Piano Concertos, Cuthbert Girdlestone. The only full-length 
study of an important area of Mozart's creativity. Provides detailed analyses of all 
23 concertos, traces inspirational sources. 417 musical examples. Second edition. 
509pp. (USO) 21271-8 Paperbound $3.50 

The Perfect Wagnerite: A Commentary on the Niblung's Ring, George 
Bernard Shaw. Brilliant and still relevant criticism in remarkable essays on 
Wagner's Ring cycle, Shaw's ideas on political and social ideology behind the 
plots, role of Leitmotifs, vocal requisites, etc. Prefaces, xxi -f 136pp. 

21707-8 Paperbound $1.50 

Don Giovanni, W. A. Mozart. Complete libretto, modern English translation; 
biographies of composer and librettist; accounts of early performances and critical 
reaction. Lavishly illustrated. All the material you need to understand and 
appreciate this great work. Dover Opera Guide and Libretto Series; translated 
and introduced by Ellen Bleiler. 92 illustrations. 209pp. 

21134-7 Paperbound $1.50 

High Fidelity Systems: A Layman's Guide, Roy F. Allison. All the basic 
information you need for setting up your own audio system: high fidelity and 
stereo record players, tape records, F.M. Connections, adjusting tone arm, cartridge, 
checking needle alignment, positioning speakers, phasing speakers, adjusting hums, 
trouble-shooting, maintenance, and similar topics. Enlarged 1965 edition. More 
than 50 charts, diagrams, photos, iv -f 91pp. 21514-8 Paperbound $1.25 

Reproduction of Sound, Edgar Villchur. Thorough coverage for laymen of 
high fidelity systems, reproducing systems in general, needles, amplifiers, preamps, 
loudspeakers, feedback,, explaining physical background. "A rare talent for making 
technicalities vividly comprehensible," R. Darrell, High Fidelity. 69 figures. 
iv-f92pp. 21515-6 Paperbound $1.00 

Hear Me Talkin' to Ya: The Story of Jazz as Told by the Men Who 
Made It, Nat Shapiro and Nat Hentoff. Louis Armstrong, Fats Waller, Jo Jones, 
Clarence Williams, Billy Holiday, Duke Ellington, Jelly Roll Morton and dozens 
of other jazz greats tell how it was in Chicago's South Side, New Orleans, depres- 
sion Harlem and the modern West Coast as jazz was born and grew, xvi -j- 429pp. 

21726-4 Paperbound $2.50 

Fables of Aesop, translated by Sir Roger L'Estrange. A reproduction of the very 
rare 1931 Paris edition; a selection of the most interesting fables, together with 50 
imaginative drawings by Alexander Calder. v + 128pp. 6 1 / 2 x9 1 / 4- 

21780-9 Paperbound $1.25 



CATALOGUE OF DOVER BOOKS 

Almost Periodic Functions, A. S. Besicovitch. Thorough summary of Bohr's 
theory of almost periodic functions citing new shorter proofs, extending the theory, 
and describing contributions of Wiener, Weyl, de la Vallee, Poussin, Stepanoff, 
Bochner and the author, xiii + 180pp. 60018-1 Paperbound $1.75 

An Introduction to The Study of Stellar Structure, S. Chandrasekhar. 
A rigorous examination, using both classical and modern mathematical methods, of 
the relationship between loss of energy, the mass, and the radius of stars in a steady 
state. 38 figures. 509pp. 60413-6 Paperbound $3.25 

Introduction to the Theory of Group's of Finite Order, Robert D. Car- 
michael. Progresses in easy steps from sets, groups, permutations, isomorphism 
through the important types of groups. No higher mathematics is necessary. 783 
exercises and problems, xiv + 447pp. 60300-8 Paperbound $3.50 

The Solubility of Nonelectrolytes, Joel H. Hildebrand and Robert L. Scott. 
Classic, pioneering work discusses in detail ideal and nonideal solutions, inter- 
molecular forces, structure of liquids, athermal mixing, hydrogen bonding, equa- 
tions describing mixtures of gases, high polymer solutions, surface phenomena, etc. 
Originally published in the American Chemical Society Monograph series. New 
authors' preface and new paper (1964). 148 figures, 88 tables, xiv + 488pp. 

61125-6 Paperbound $3.00 

Introduction to Applied Mathematics, Francis D. Murnaghan. Introduction 
to advanced mathematical techniques — vector and matrix analysis, partial differen- 
tial equations, integral equations, Laplace transform theory, Fourier series, 
boundary- value problems, etc. — particularly useful to physicists and engineers. 4l 
figures, ix + 389pp. 61042-X Paperbound $2.25 

Elementary Mathematics From An Advanced Standpoint: Volume I — 
Arithmetic, Algebra, Analysis, Felix Klein. Second-level approach, illumi- 
nated by graphical and geometrical interpretation. Covers natural and complex 
numbers, real equations with real unknowns, equations in the field of complex 
quantities, logarithmic and exponential functions, goniometric functions, infini- 
tesimal calculus, transcendence of e and ir. Concept of function introduced im- 
mediately. Translated by E. R. Hedrick and C. A. Noble. 125 figures, ix + 274pp. 

(USO) 60150-1 Paperbound $2.25 

Elementary Mathematics From An Advanced Standpoint: Volume II — 
Geometry, Feliex Klein. Using analytical formulas, Klein clarifies the precise 
formulation of geometric facts in chapters on manifolds, geometric and higher 
point transformations, foundations. "Nothing comparable," Mathematics Teacher. 
Translated by E. R. Hedrick and C. A. Noble. 141 figures, ix -f 214pp. 

(USO) 60151-X Paperbound $2.25 

Engineering Mathematics, Kenneth S. Miller. Most useful mathematical tech- 
niques for graduate students in engineering, physics, covering linear differential 
equations, series, random functions, integrals, Fourier series, Laplace transform, 
network theory, etc. "Sound and teachable," Science. 89 figures, xii + 4l7pp. 
6x8l/ 2 . 61121-3 Paperbound $3.00 



CATALOGUE OF DOVER BOOKS 

Introduction to Astrophysics: The Stars, Jean Dufay. Best guide to ob- 
servational astrophysics in English. Bridges the gap between elementary populariza- 
tions and advanced technical monographs. Covers stellar photometry, stellar spectra 
and classification, Hertzsprung-Russell diagrams, Yerkes 2-dimensional classifica- 
tion, temperatures, diameters, masses and densities, evolution of the stars. Trans- 
lated by Owen Gingerich. 51 figures, 11 tables, xii -f 164pp. 

(USCO) 60771-2 Paperbound $2.00 

Introduction to Bessel Functions, Frank Bowman. Full, clear introduction to 
properties and applications of Bessel functions. Covers Bessel functions of zero 
order, of any order; definite integrals ; asymptotic expansions ; Bessel' s solution to 
Kepler's problem; circular membranes ; etc. Math above calculus and fundamentals 
of differential equations developed within text. 636 problems. 28 figures, x + 
135pp. 60462-4 Paperbound $1.75 

Differential and Integral Calculus, Philip Franklin. A full and basic intro- 
duction, textbook for a two- or three-semester course, or self-study. Covers para- 
metric functions, force components in polar coordinates, Duhamel's theorem, 
methods and applications of integration, infinite series, Taylor's series, vectors and 
surfaces in space, etc. Exercises follow each chapter with full solutions at back 
of the book. Index, xi + 679pp. 62520-6 Paperbound $4.00 

The Exact Sciences in Antiquity, O. Neugebauer. Modern overview chiefly 
of mathematics and astronomy as developed by the Egyptians and Babylonians. 
Reveals startling advancement of Babylonian mathematics (tables for numerical 
computations, quadratic equations with two unknowns, implications that Pytha- 
gorean theorem was known 1000 years before Pythagoras), and sophisticated 
astronomy based on competent mathematics. Also covers transmission of this 
knowledge to Hellenistic world. 14 plates, 52 figures, xvii + 240pp. 

22332-9 Paperbound $2.50 

The Thirteen Books of Euclid's Elements, translated with introduction and 
commentary by Sir Thomas Heath. Unabridged republication of definitive edition 
based on the text of Heiberg. Translator's notes discuss textual and linguistic 
matters, mathematical analysis, 2500 years of critical commentary on the Elements. 
Do not confuse with abridged school editions. Total of xvii + 1414pp. 

60088-2, 60089-0, 60090-4 Three volumes, Paperbound $8.50 

An Introduction to Symbolic Logic, Susanne K. Langer. Well-known intro- 
duction, popular among readers with elementary mathematical background. Starts 
with simple symbols and conventions and teaches Boole-Schroeder and Russell- 
Whitehead systems. 367pp. 60164-1 Paperbound $2.25 



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Guide to the Literature of Mathematics and Physics, 

Nathan G. Parke III. (60447-0) $3.00 
A Survey or Physical Theory, Max Planck. (60650-3) $1.50 
The Theory of Heat Radiation, Max Planck. (60546-9) $2.50 
Treatise on Thermodynamics, Max Planck, (60219-2) $2.50 
Scientific Papers, Lord Rayleigh (John William Strutt). 

(61213-9, 61214-7. 61215-5) Three-volume set, Cloth- 
bound $37.50 
The Theory of Sound, Lord Rayleigh (John William Strutt). 

(60292-3, 60293-1) Two-volume set $6.00 
Radiochemistry and the Discovery of Isotopes, edited by 

Alfred Romer. (62507-9) $3.50 
The Discovery of Radioactivity and Transmutation, edited 

by Alfred Romer. (61198-1) $2.50 
Selected Papers on Cosmic Ray Origin Theories, edited by 

Stephen Rosen. (62127-8) $5.00 
History of Hydraulics, Hunter Rouse and Simon Ince. 

(61131-0) $2.50 
The Analysis of Matter, Bertrand Russell. (60231-1) $3.00 
Essay on the Foundations of Geometry, Bertrand Russell. 

(60233-8) $2.50 
The Study of the History oi Mathematics and The Study 

of the History of Science, George Sarton. (20240-2) 

$2.25 
History of Mathematics, David Eugene Smith. (20429-4, 

20430-8) Two-volume set $8.00 

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THE HISTORY OF 

THE CALCULUS AND ITS 

CONCEPTUAL DEVELOPMENT 

CARL B. BOYER 



This book., for the first time., provides laymen and mathematicians alike with a detailed 
picture of the historical development of one of the most momentous achievements of the 
human intellect— the calculus. It describes with accuracy and perspective the long 
development of both the integral and the differential calculus from their early beginnings 
in antiquity to their final emancipation in the 19th century from both physical and 
metaphysical ideas alike and their final elaboration as mathematical abstractions, as we 
know them today, defined in terms of formal logic by means of the idea of a limit of 
an infinite sequence. ^ 

But while the importance of the calculus and mathematical analysis — the core of modern 
mathematics cannot be overemphasized, the value of this first comprehensive critical 
history of the calculus goes far beyond the subject matter. This book will fully counteract 
the impression of laymen, and of many mathematicians, that the great achievements of 
mathematics were formulated from the beginning in final form. It will give readers a 
sense of mathematics not as a technique, but as a habit of mind, and serve to bridge the 
gap between the sciences and the humanities, It will also make abundantly clear the 
modern understanding of mathematics by showing in detail how the concepts of the 
calculus gradually changed from the Greek view of the reality and immanence of 
mathematics to the revised concept of mathematical rigor developed by the great 19th 
century mathematicians, which held that any premises were valid so long as they were 
consistent with the one another, It will make clear the ideas contributed by Zeno, Plato, 
Pythagoras, Eudoxus, the Arabic and Scholastic mathematicians, Newton, Leibnitz, Taylor, 
Descartes, Euier, Lagrange, Cantor, Weierstrass and many others in the long passage 
from the Greek "method of exhaustion" and Zeno's paradoxes to the modern concept of 
the limit independent of sense experience; and illuminate not only the methods of 
mathematical discovery, but the foundations of mathematical thought as we 

Complete, unabridged republication. Foreword by R. Courant. Preface. 22 figures. 25-page 
bibliography. Index, v + 346pp. %% x 8. 60509-4 Paperbound 



A DOVER EDITION DESIGNED FDR YEARS OF USE! 

We have made every effort to make this the best hook possible. Our paper is opaque, with 
minimal show-through; it will not discolor or become brittle with age. Pages are sewn in 
signatures, in the method traditionally used for the best books, and will not drop out as often 
happens with paperbacks held together with glue. Books open flat for easy reference. The 
binding will not crack or split. This is a permanent book. 



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